lasertweezers report van gorkom

8/10/2019 LaserTweezers Report Van Gorkom

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Optical TweezersKyle Van Gorkom

Physics 39aSeth Fraden

13 December 2013

Abstract

Optical tweezers allow for the manipulation of microscopic objects by means of afocused laser. In this lab, a simple optical tweezers setup is constructed. The trapping

behavior of the laser is explained in terms of dielectric dipole attraction and geometricoptics. The potential well set up by the optical tweezers is characterized in terms of itsshape, size, and the strength of its restoring force through two experimental setups.Finally, a third experimental setup creates a double potential well and characterizes thisdouble well in terms of its shape.


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1. Theory

1.1 Trapping Theory

Optical tweezers operate by bringing a

laser beam into focus in the plane of amicroscope sample. This focused beam canthen be used to trap and manipulatemicroscopic objects. Two approaches areadopted in explaining the physics behindoptical tweezerselectrodynamics andgeometric optics.

Under the first regime, we consider ourfocused laser as an electromagnetic wave thatsets up an electric field in the sample plane. Ifthe object to be trapped is a dielectric material,

this will induce a dipole moment proportionalto this electric field, = . Then the

potential is given by = = 2.The laser intensity can be more directly measured and controlled and is easily related tothe electric field by the relation 2. Thus, . From this expression, therelationship between force and the intensity is easily found. We know = , so

. Thus, a dielectric in the vicinity of thelaser focal point will feel an attractive force in thedirection of the greatest intensity.

The above reasoning, however, requiresthat we be able to treat our object to be trapped asa perfect dipole, which would only be reasonablein the limit that the object radius is very smallcompared to the wavelength of the laser, or

. In the opposite regime, , we canadopt a geometric optics approach.

If we limit ourselves to considering thewidest angle rays that make up the convergingfocal cone of the laser, as in Figure 2 (where theincident rays are 1 and 2), we can explain thetrapping in terms of momentum imparted to theobject by the reflected and refracted rays. In thecase illustrated, the reflected rays 11 and22 push the sphere downwards, while the

refracted rays 12and 22 push the sphere upwards. (By symmetry, the horizontal forcescancel and can be neglected in this analysis). If the force due to the reflected wavesdominates, then the sphere will be accelerated downwards out of the trap; if, however, theforce due to the refracted waves dominates, the sphere will be pulled up into the beam,creating a stable trap.

Figure 1: A dielectric attracted into an electricfield.

Figure 2: Optical rays incident on a sphere.


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In our experimental setup, the beads (the sphere in Figure 2) have a radius ofapproximately 1 m, while red laser light has a wavelength on the order of 0.65 m.Thus, neither the dipole attraction nor the geometric optics regime holds exactly for ourcase. However, the two approaches are illustrative of the physical mechanism behind thetrapping force created by the laser.

1.2 Characterizing the Potential Well

When framed in termsof potentials, the trap can beunderstood as a well, intowhich the dielectric bead falls.Since it will be a minimum ofthe potential, it can beexpanded and, to the first order,expressed as

( ) = 12

2

where k is the trap stiffness. Inreality, the well will approachsome constant as andwill look closer to the blackline in Figure 3 (where the redline represents the quadratic potentialof Equation (1)). We expect, then,that Equation (1) will be a reasonable approximation close to the minimum but will breakdown as we approach the edge of the trap.

By equipartition theorem,< ( ) > =

12

where is the Boltzmann constant and T the temperature. Equating Equation (1) and (2)and solving for trap stiffness , we find:

=< 2 >

For a trap centered at 0, the denominator becomes < ( 0)2 > . For adistribution of measured positions, we take 0to be the mean position of the bead; thus,finding the variance of a trapped particle will allow us to calculate the trap stiffness.

The Boltzmann distribution gives us another approach to characterizing the well,

without making any assumptions about its shape. Let ( ) be the distribution ofmeasured positions of a particle in a well. Then

( ) = ( )/

where Z is the partition function. Solving for ( ) gives:

( ) = ( ) + where Z has been absorbed into the constant and can be neglected by setting = 0. In

U(x)

Figure 3: The potential well setup by the optical tweezers.The red line marks the parabolic approximation.

(1)

(2)

(3)

(4)

(5)


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principle, Equation (5)could be used to map outthe extremes of the wellwhere the quadraticapproximation fails.

Unfortunately, a bead in astationary trap will tend tostay near the minimum;thus, we require anothermethod of characterizingthe well extremes.

The trap well canalso be understood in termsof the restoring force itexerts, since = U.This is plotted in Figure 4,

where the black line represents thereal force and the red line theforce of the quadratic approximation.

If we give the trap some velocity, the trapped bead will feel a drag force. In theregime of laminar free flow, this will be given by the Stokes equation:

= 6 where is the fluid viscosity, the bead radius, and the velocity. This drag force is

plotted on Figure 4 as the dotted line, when increased to the point at which it exceeds therestoring force and pulls the bead out of the trapthe inflection point of the trap

potential. This gives us a means of finding the outer extreme of the trap, .

By comparing the intersect of the actual trapping force with the drag force againstthe same intersect for the trapping force under the quadratic approximation, we expect tosystematically underestimate under this approximation. Nonetheless, equating

and = , we find=

6

On top of the systematic underestimation due to our approximation, we will havean underestimation of the calculated drag force, as Equation (6) assumes free flow. Inreality, our bead may be trapped close to the upper slide of the sample. This could lead toan underestimation of the calculated value of by a factor of 10.

2. Apparatus and Procedure

2.1 Optics Setup

The optical setup employed for all experiments is pictured in Figure 5, where the path of the laser is traced in red. The primary elements consist of an intensity sensor, a pair of movable mirrors driven by motors to steer the laser, a pair of 17cm planoconvexlenses setup in a 1-to-1 imaging arrangement, a 4x magnifying lens, a dichroic mirror,

x

F(x)

FD(v)

Figure 4: The restoring force setup by the optical

tweezers. The red line marks the parabolic approximation.

(6)

(7)


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and a camera. The dichroic mirror redirects the laser while allowing light from thesample image to reach the camera, which was connected via USB to a computer. Thesample was a collection of micron-scale dielectric beads suspended in water.

Three primary considerations drove the configuration of the optics. First, the lasermust come to a focus on the sample. Second, the back focal plane of the objective must

be filled by the laser to produce high angle rays that converge on the sample. Third, asthe laser is steered by the movable mirrors, the laser must always enter the opening of theobjective.

2.2 Image Processing

The images captured by the mounted camera wererecorded in AMCAP and processed in ImageJ. In order tofilter out the image of the laser, the green channel wasextracted from the captured videos and used for all image

processing. After cropping, a brightness threshold wasset to pick out the center of the bead under consideration(Figure 6), and a minimal particle area set to avoidtracking noise or other features. The exact thresholdand minimal area had to be adjusted for each video in

Figure 5: The optical setup for the laser tweezers. The path of the laser is markedin red, and the path of the microscope illumination is marked in blue.

Figure 6: A close-up of a trappedbead and the detection mask used totrack its center.


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order to produce optimal results. In this manner, the center of the bead was picked outfrom each frame, and a list of bead centers constructed. Depending on the particularexperiment, recorded data ranged from 30 to 90 seconds, or approximately 500 to 1300frames.

2.3 Experimental ProcedureThree separate experiments were carried out, with the respective goals of

characterizing the bottom of the laser trap, the extremes of the trap, and to create aclosely-spaced double well, as well as to observe the dependence of the well features onlaser intensity.

To characterize the bottom of the potential well, a bead was captured in astationary trap. The motion of the particle in the trap was extracted from recordingsmade, and these data used to calculated the trap stiffness (Equation (3)) and the shape of

potential under the Boltzmann distribution (Equation (5)). This was repeated at a varietyof laser intensities. These data were collected twiceonce in which the same particle was

used across all intensities, and once in which a different particle was selected at eachintensity.In the second experiment, a particle was trapped and the laser steered by a triangle

wave of known amplitude. The frequency of the triangle wave was increased until the bead fell out of the trap. Figure 7 shows the horizontal position of the particle againsttime, until the moment right before the particle exited the well. This gives us themaximum trapping force of the well and can be used to estimate the width of the well byEquation 7. To account for the error introduced from switching beads throughout theexperiment, the critical velocity was found for four separate beads at each laser intensity.

Finally, a high frequency (about 40 Hz) square wave was used to steer the welland to create two adjacent wells. To capture the particle, the amplitude of the wave wasset to zero. Once a bead was captured, the amplitude was increased (to a maximum ofapproximately 1 micron) until the bead jumped back and forth between the two wellminima. Recordings of the bead yielded position distributions that were, as in the firstexperiment, used to map out the shape of the potential.

Figure 7: The position of the center of a bead trapped in a well driven by a trianglewave of increasing frequency.


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3. Data and Results

3.1 Well Minima

In the stationary trap, the distribution of positions for a single bead at a given laserintensity is approximately Gaussian. The histogram for two relative laser intensities 0.15 and 0.9are shown in Figure 8. As expected, lower laser intensities show a broaderdistribution, while higher intensities show a narrower distribution.

Two approaches to characterizing the shape of the potential for the distributions inFigure 8 are shown in Figure 9. The individual points are calculated from Equation (5)under the assumption of a Boltzmann distribution. The red line is the quadratic potentialwith a spring constant calculated from Equation (3). The two approaches yield verysimilar results, lending support to the accuracy of the quadratic assumption at the bottomof the well. Figure 10 plots the 2-dimensional Boltzmann potential for the 0.15 intensitycase.

Finally, we can characterize more explicitly the dependence of the trappingstiffness on laser intensity by plotting the spring constant against laser intensity. Figure11 shows the results for data collected from the same dielectric bead with a line fit to thedata in MATLAB, given by 1.1 10 5 6.3 10 7, where is the laser intensity.Figure 12 shows the results when a different bead was selected at each intensity. Thelatter shows greater scatter, suggesting that differences among the beads played some rolein determining the results.

Figure 8: Distribution of trapped bead positions for relative laser intensities of 0.15 (left) and 0.9(right).

Figure 9: Boltzmann potential (blue) against quadratic potential (red) calculated from thedistributions in Figure 8.


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Figure 10: The 2-dimensional shape of the 0.15 intensity potential well. The potential is given inarbitrary units.

Figure 11: Trap stiffness (left) and (right) plotted against laser intensity for a single particle.

Figure 12: Trap stiffness (left) and (right) plotted against laser intensity for a different particleat each intensity.


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3.2 Well Extremes

The maximum trapping force was calculated by Equation (6). Figure 13 plots thecalculated force against intensity, where each point is the mean of four separatemeasurements and the error bars represent one standard deviation. As expected, higher

laser intensities correlate with a higher trapping force.

We plot this force, against the spring constant result from the stationarytrap in Figure 14. Then = ( ) = . So the slope in Figure 14should give us the well boundary. The line fit gives a value of approximately 0.1 m.However, this result assumes Stokes drag in a regime of free flow. Accounting for the

sample wall could increase the calculated drag force by a factor of 10, giving us 1 . Doing the explicit calculation for from Equation (7) gives the result shown inFigure 15. Calculated values of by this method range from 0.03 to 0.16 m.

Figure 13: Maximum trapping force plotted against laser intensity.

Figure 14: Maximum trapping force against trap stiffness .


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3.3 Double Well

Figure 16 shows the measured position distribution of a double well. For thisresult, the laser intensity was set to 0.5, and the two laser focal points were separated byapproximately 1 m. The axis along the double well shows a roughly bimodaldistribution, while the axis perpendicular to the double well appears approximatelyGaussian, as in the case of the stationary well. An asymmetry existed between the twowells, with the bead preferring to fall into the well on the left. This asymmetry manifests

itself clearly in the Boltzmann result for the potential in Figure 17the well on the left ismuch deeper than the well on the right. Once again, along the axis perpendicular to thedouble well, the potential appears parabolic. Finally, the 2-dimensional Boltzmann

potential is shown in Figure 18.

Figure 15: Calculated values of the well boundary over a range of

laser intensities.

Figure 16: Distribution of trapped bead positions along the axis of the double well (left) andperpendicular to the double well (right).


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4. Conclusion

The construction of an optical trap capable of manipulating micron-scale beadswas successfully demonstrated. By comparison with the results of the Boltzmanndistribution, the assumption of a quadratic potential near the well minimum was shown to

be a reasonable one. As expected, trap stiffness, as a measure of the strength of therestoring force of the trap, increased with the intensity of the laser. A double well wasconvincingly made, although never without an asymmetry resulting in the bead preferringone well over the other. A rough estimate of the trap size was possible, but depended on anumber of assumptions known to not quite hold. It would be interesting to calculate thediffusion constant for a trapped bead in our sample and use that to obtain a more accuratemeasure of the well size.

Figure 17: Potential shapes calculated from the Boltzmann distribution along the axis of the doublewell (left) and perpendicular to the double well (right).

Figure 18: The 2-dimensional Boltzmann potential for the double well. Potential and distances aregiven in arbitrary units and scaled arbitrarily.


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Sources Consulted

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D. C. Appleyard et al., Optical trapping for undergraduates, Am. J. Phys. (1), 5-14

(2007)A. Ashkin, Forces of a single-beam gradient laser trap on a dielectric sphere in the ray

optics regime, Biophys. J. (61), 569-572 (1992)J. Bechhoefer and S. Wilson, Faster, cheaper, and safer optical tweezers for the

undergraduate laboratory, Am. J. Phys. 70 (4), 393-400 (2002)S. Smith, et al. Inexpensive optical tweezers for undergraduate laboratories, Am. J.

Phys. 67, 26-35 (1999)M. Williams, Optical Tweezers: Measuring Piconewton Forces

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