Measurement of Trap Stiffness of a Single Beam Optical Tweezers

Measurement of trap stiffness of a single beam optical tweezers

Rolf Karlstad Lab partner: Joe Peterson University of Minnesota School of Physics and Astronomy Submitted May 5, 2006

We construct a single beam optical tweezers and trap 6.02 micron, n=1.57 polystyrene micro- spheres. Using a CCD camera and imaging software we determine the trap stiffness by measuring the change in the position of the sphere caused by the viscous drag force of water flowing past the sphere. We measured the trap stiffness of our single-beam Optical Tweezers to be 3.94±0.09 piconewtons/micron, with a maximum trapping force of 7.90 piconewtons.

Introduction Optical tweezers are a recent development in experimental physics. The idea was not implemented until the mid-80’s, and an original designer, Steven Chu, earned a Nobel Prize in Physics in 1997 as a result of his research. Optical Tweezers are currently being used to study DNA and RNA interactions and have been able to measure femtoNewton (10-15N) forces of molecular motors[a].

A laser beam properly focused on a polystyrene sphere suspended in water will cause the sphere to become trapped. The optical trapping force is due to the change in momentum of photons as they pass through the sphere. As rays from the focused beam cross the interface between the water and sphere, they are refracted according to Snell’s Law: (1) nsinq= n sin q water incident sphere refracted

Here θincident is the angle between a ray incident upon the sphere and the line normal to the sphere at this point, and θrefracted is the angle between the refracted ray and the extension of the normal to the interior of the sphere. This causes a change in the momentum of the photons which comprise the beam. An equal but opposite change in momentum must then be experienced by the sphere due to the conservation of momentum. The resulting force of two rays emanating from opposite edges of the beam points towards the lens focusing the beam. This force is referred to as the refractive force, and is the force that causes the trap (see Figure 1). The other force of importance is the force due to reflection at the bead surface, illustrated in Figure 1. Depending on how much light is reflected at the bead surface, the force due to reflection can easily push the sphere out of a weak single-laser trap. In order to minimize the force of reflection, a microscope objective with a high numerical aperture must be used for a single beam Tweezers. Further, trap stiffness is proportional to both laser power and objective numerical aperture, further illuminating the importance of high numerical aperture. The maximum numerical aperture for a given microscope objective is constrained by the index of refraction of the medium following the objective. This means that an oil-immersion objective is necessary to maximize trapping force for a given laser power, since oils used for this purpose have indices of refraction up to 1.4+, where air has index of refraction 1.0.

In this experiment, we determine the trap stiffness of an optical tweezers by measuring the change in position of a trapped sphere when experiencing measured viscous drag forces.

Figure 1: Diagram showing the force on a dielectric sphere due to reflection and refraction of two rays of light, 1 and 2. The resultant force on the bead due to refraction is towards the focus. [Reference b, Williams]

Theory For this experiment the optical tweezers is created by a laser beam, expanded by a pair of lenses serving as a beam expander, whose diameter matches the ‘input’ aperture of a microscope objective which focuses the beam into a cell of water containing polystyrene spheres. Water is allowed to flow from an upper reservoir, through the cell and into a lower reservoir. Spheres are injected as needed, via a syringe inserted into the water flow tubing. In order for the sphere to be trapped by the laser, the index of refraction of the sphere must be greater than that of the surrounding water. If this were not the case, we see from (1) that the light would be bent upwards relative to the normal line and the resulting force would push the sphere away from the trap. We also need Frefractive>Freflective, which is satisfied by our choice of polystyrene spheres and high numerical aperture microscope objective. The key to the trap is that the intensity of the beam is a maximum at its center and decreases as you move from the central maximum. With no outside forces acting on the sphere, it will remain trapped in equilibrium in the center of the trap. If any force pushes it away from this equilibrium, the trap will create a restorative force due to the intensity differences of rays from the beam (see Fig. 2).

Figure 2: Restorative force on sphere due to intensity profile [Reference c, Dholakia, Spalding, MacDonald, 2002.]

The cause is the difference in the magnitude of refractive forces due to the difference in ray intensity. Although the total force acting on the sphere is the sum of the forces over the entire beam, it is easier to visualize by looking at just the outermost rays. As the sphere moves to either side, the outside ray acting on the sphere has a smaller intensity than the inner ray. In this case the resulting force pushes the sphere back to the center. Thus, any perturbation of the sphere from center will result in a force restoring it to the center, so the sphere is trapped in a potential well. However, a large enough force will knock the sphere out of the trap. The restorative force of the trap on a sphere can be modeled by:  F  kx (2) With the position of the sphere, x, measured from the center of the trap. The center is given by the position of a sphere trapped when no other forces are applied, and is at the center of the focus of the beam. By measuring the position changes of a trapped sphere when we apply known forces, we can use linear least-squares fitting to determine k, the trap stiffness. The position is measured with a CCD camera, forces are provided by water flowing past the sphere. The viscous drag force produced by water flowing at velocity v, past a sphere with radius r is: (3) F= 6ph rv drag Here η is the viscosity of water, which is temperature dependent. In order to produce known forces we need to be able to determine the velocity of the water as it flows through the cell. To measure velocity of the water flow, we released our trapped beads after gathering position data under flow. By measuring the position of the released bead, we ensured that velocity was measured coplanar to our initial measurements. We controlled water velocity through the cell by clamping a water flow line with a optical table clamp. By varying the tension on the clamp, we changed the cross-sectional area of the tubing. This provided a range of relatively random velocity values. Velocity data is sufficient to determine Fdrag in (2). Now we can produce forces of known magnitude on a sphere in the trap and obtain k by (2).

Experimental Apparatus and Technique Mirror Laser

40 mm lens

(Beam expander)

200 mm lens Cell CCD White Light Source

Objective Color Filter Laser line mirror Tip

Figure 3: Schematic Layout of Optical Tweezers. [Reference e, Williams] Our experimental setup is similar to that shown in Figure 3. Additionally, the camera is attached to an eyepiece, which provides a field correction and mounting. Our camera is a Philips Toucam 740k, with the objective lens removed. This is a USB webcam with 740K pixels. It is attached with tape to a Pentax XP 14mm telescope eyepiece. A 20mW, 633 nm HeNe laser is used to produce the light for the trap. The laser beam is first reflected off a 88% reflective aluminum mirror. This mirror allowed convenient laser placement. The reflected beam is then sent through two lenses in order to enlarge the diameter of the beam to match the size of the internal aperture of the microscope objective; this helps to maximize the trapping force available. Our lenses were 200mm and 38mm, respectively, and expanded the laser beam by a factor of 5. The expanded beam is then reflected off a dichroic mirror that reflects light about a narrow band of frequencies and passes all other frequencies. This mirror reflects the 633 nm light of the laser into the microscope objective, but passes shorter and longer wavelengths. An oil immersion objective is used to focus the beam into the flow cell. In order to observe what is occurring in the cell, a CCD camera is placed behind the dichroic mirror and looks into the flow cell through the eyepiece and objective. Between the eyepiece and the objective is another lens, which adjusts the image formed on the CCD to be coplanar with the focused laser light. Since the laser light is collimated, but we are using a microscope objective with a fixed ‘tube length’, we must make this accommodation if we hope to visualize the plane onto which the laser is focused. A white light source illuminates the beads from the opposite side of the flow cell. We placed a cylinder filled with water into the white light beam in an effort to reduce IR and UV emissions. In order to analyze the bead movement we employ a color filter that filters out light around the laser’s frequency range, so that laser light does not interfere with our observation of the beads. The camera interfaces to the computer over a standard USB interface, and IRIS software directly records movies from the camera to uncompressed AVI files, which are then broken down into individual FITS images.

The flow cell design is seen in figure 4. Water flows from an upper reservoir through the cell and is collected in a lower reservoir. The flow cell is constructed from a plastic spacer glued with epoxy between two #1 glass cover slips. A schematic is found in the appendix. Water flows in and out of the cell through metal tubing that is inserted and glued with epoxy into the 1/16” holes drilled in the side of the spacer. Figure 4-Design of reservoirs and flow cell for determining viscous drag forces [Reference b, Williams] The reservoirs used are calibrated fluid containers, and are connected to the cell by Tygon tubing with a 1/16” internal diameter. Additionally, a length of 1mm ID Teflon tubing was used in an effort to reduce flow rates. These reservoirs were chosen to be much wider than they are deep, such that the gradient force due to change in height remains relatively constant during a data capture run. Polystyrene beads are injected into the flow cell by a syringe stuck into the tubing leading from the upper reservoir to the cell. The upper reservoir is attached to a stand that allows us to adjust its height to provide a coarse range of flow rates, the table clamp deforming the outlet tube controls the actual rate of flow. We used Tygon and Teflon tubing and Corning snap-close reservoirs. The 6.02 micron diameter micro-spheres are manufactured by Bangs Laboratories, Inc. I personally supplied a Leitz-Wetzlar 1.3 NA 100x microscope objective, Pentax 14mm XP eyepiece, and Philips Toucam Pro 740K, along with a light condenser which was helpful for matching the high NA of the 100x objective.

I personally constructed several flow cells that did not work. The primary lesson from these flow cells is that the cross-section should be relatively square, and should be on order of 3x3mm in cross-section. Also, epoxy should be used, rather than attempting any other mechanical sealant such as Parafilm. We also began our project using an MXP 56x, 0.95 NA microscope objective. This is not an appropriate objective for this project. An inexpensive objective such as the Edmund 1.25 NA, 100x objective would be a much better choice. A higher power laser would be very desirable, and I plan to experiment over the summer to determine whether or not an inexpensive diode or other type laser might be able to replace the much more expensive HeNe laser used in this experiment. Data Analysis IRIS imaging software [f] proved an excellent tool to analyze videos made of beads in the flow cell. By highlighting the bead in a frame of the video, IRIS tracks the bead as it changes positions in each frame of the video. A subroutine determines the pixel shifts of the selected section relative to the position in the initial frame and outputs these shifts to a text file. Since the beads used are of known diameter and we can measure their diameters in pixels or even sub pixels, we can directly convert between pixels and microns. If flow is introduced into the cell, a bead traveling freely through the cell will travel at the same velocity as the water. Hence, a video recorded from the CCD camera of a polystyrene bead traveling though the cell will allow us to determine its velocity of the water. We measure the velocity of the water as it travels through the flow cell by measuring the change in position of a once-trapped bead, released and flowing with the water. Combining this with the rate of frame capture, we determine the average velocity of the bead between frames. The velocity of flow through the cell is determined for each data run by releasing the previously trapped bead and measuring position vs. time during a period of zero acceleration.

Once water velocity is determined, the viscous drag force as a function of height is obtained from (3) by measuring the temperature of the water to determine its viscosity[d]. A sphere is trapped, as confirmed by video. Water flow is then induced by opening the flow valve. The position of the trapped bead in response to a given external force (water velocity) is measured. By using the same techniques used to determine the water velocity, we measure the position shifts of the bead from the center of the trap under known forces with IRIS. We obtain the standard deviation of the position by the excel function STDEV. We were able to calibrate our distance to pixel scale as 14.5±0.5 pixels per micron. This uncertainty in distance measurement, followed by a 2.5% uncertainty in water viscosity, was the largest source of error in our results. By using linear least-squares fitting of displacement, viscosity, and velocity data to (2), we arrive at our desired result: the trap stiffness k. Results We gathered data for a range of water velocities. 10 second movies were recorded at 30 frames per second. The movies contained images of a bead trapped in the equilibrium position with no water flow through the cell, followed by the bead trapped and displaced from equilibrium while water flowed through the cell, and finally the bead flowing along with the water after the laser beam was interrupted. A composite graph of position vs. time for the trapped beads follows. Position vs. Time -- Trapped at equilibrium, then under flow

25 ) s l e x i p

( 20 run I n o i t run F c

e 15 l f

e run E d 10 run D run C 5 run B run A 0 0 1 2 3 4 5 6 -5 time (seconds)

Figure 5 – Position vs. time of beads trapped at equilibrium, then under flow.

The release data at the end of each movie needed to be analyzed separately. A graph of ‘terminal’ velocity in the water flow follows for the different runs.

Determination of water Velocity by bead release in flow (Released bead position vs. time: sampling began when delta V = 0)

200 ) s l e x i P (

n 150 o i t i s o P

100 run f run i run e run d 50 run c run b run a

0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

-50 Time (seconds)

Figure 6 – Position vs. time of beads released under flow, determining water velocity Error propagation for this experiment consisted originally of determining the uncertainty in bead position. This was found to be 0.5 micron by the standard deviation of bead position in equilibrium position in the trap. This error propagated into the LSQ spreadsheet, written by Kurt Wick [g], to provide an uncertainty in the velocity of the water flow. Uncertainty in viscosity was determined by difference in temperatures in the two reservoirs. Our primary force equation is of the form F(a, b, c) = a*b*c, so the method of propagating partial errors was adequate. Specifically, the fractional uncertainty in F is equal to the fractional uncertainty in a, b, and c, added in quadrature. All errors were propagated in this manner.

Our position and velocity data was compiled in the following graph plotting force vs. displacement.

F o r c e v s . D is p l a c e m e n t

6E-12 ) s

n 5E-12 o t w e N (

e 4E-12 c r o F

2E-12 y yfit 1E-12

0 0.00E+00 2.00E-07 4.00E-07 6.00E-07 8.00E-07 1.00E-06 1.20E-06 1.40E-06 1.60E-06 1.80E-06 2.00E-06 Distance (meters)

Figure 6 – Force vs. Displacement of beads trapped under flow, determining trap stiffness.

From the graph in figure 6, we determined the trap stiffness of our optical tweezers to be 3.94±0.09 piconewtons/micron. The maximum trapping force measured was 7.90 pN. These values and their uncertainty were determined by the LSQFit spreadsheet. Our χ2 value was 7.8 for 5 degrees of freedom, for a reduced χ2 of 1.56. A graph of χi follows.

G raph of i for each data point

0 1 2 3 4 5 6 7 8

Acknowledgments

I wish to extend thanks to everybody involved in this project: My advisor Kurt Wick for his sense of humor and trusting me with being able to conduct many aspects of this experiment on my own, my lab partner Joe Petersen for signing up on this project and helping me with grunt work. Ultimate thanks, however, go to Mark C. Williams of Northeastern University. Without Mark I would not have heard about Optical Tweezers many years ago. I would not have considered this project as straightforward as it was. Mark provided invaluable assistance, in the form of email advice and documents detailing the specifics of how to construct a single-beam optical tweezers. I made every attempt to design my own apparatus, but kept going back to Mark’s original suggestions again and again. Also thanks to Paul Crowell for the encouragement and safety advice. Finally, thanks to all my fellow MXP students for stopping in and providing moral support. My most significant contribution was my time spent in the lab and my intimate knowledge of the IRIS software. I am not aware of Freeware such as IRIS being used to measure piconewton forces in the past. Which brings me to my final thank you: Christian Buil, author of the freeware Astronomical image processing software IRIS. I cannot begin to imagine how I would have analyzed the data for this project with such precision and ease, without IRIS. References

[a] K. Dholakia, P. Reece. Optical micromanipulation takes hold. Nano Today, Volume 1, Number 1. February 2006.

[b] M. C. Williams. Optical Tweezers: Measuring Piconewton Forces. Previously published in Biophysics Textbook Online. Available at: http://www.biophysics.org/education/williams.pdf

[c] K. Dholakia, G. Spalding, M. MacDonald. Optical tweezers: the next generation. Physics World, October 2002.

[d] Handbook of Chemistry and Physics, 80th edition. CRC Press, Florida. Pg 6-3. 1999.

[e] M. C. Williams. Northeastern University, Boston, MA. http://www.dac.neu.edu/mark/APL/Optical%20Tweezers.pdf

[f] see http://www.astrosurf.com/buil/us/iris/iris.htm for more information on IRIS. (Freeware)

[g] see http://mxp.physics.umn.edu/resources/images/LSQFit.xls to find LSQFit. Appendix

[Reference e, Williams]

Bangs Laboratories: Manufacturers of micro-spheres:

http://www.bangslabs.com/company/index.php Bangs Laboratories, Inc. Technology Drive Fishers, IN 46038-2886

Telephone: 317-570-7020 Fax: 317-570-7034 Toll-Free: 800-387-0672 e-mail:[email protected]