kevin george thesis final draft

8/8/2019 Kevin George Thesis Final Draft

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1.0 Abstract

An optical trap is a device which utilizes a highly focused laser in order to apply

piconewton forces to micron and nanoscale dielectric objects so as to constrain them to

the equilibrium position of the trap. Momentum changes in the incident photons give rise

to restorative forces on the trapped object. Laboratory supplies and a small budget were

utilized to design and construct an optical trap. A 120 mW, 780 nm diode laser, 100x

microscope objective lens, and a number of optical elements were used to construct a

fully functioning optical trap.

Tests were conducted to optimize and characterize the trap. It was determined

that a maximum trapping strength of 3.1 pN was achieved. Additionally, the unexpected

characteristics of trap strength were attributed to the poor beam mode quality. With poor

beam mode quality, increases in beam power at the microscope objective were the most

important factor in increasing the trap strength.


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2.0 Introduction

2.1 Introduction-Light

Light is electromagnetic radiation, the manner by which our eyes observe the

world around us. However, light is more than the just visible spectrum, the miniscule

portion of the broader electromagnetic spectrum that our eyes can see, from

approximately 400 to 800 nanometers. X-rays, ultraviolet, infrared, microwaves,

television, radio, and terahertz waves are other regions of the electromagnetic spectrum

defined by wavelength. Each region of the electromagnetic spectrum has special

properties that make it applicable for a variety of purposes. For example, x-rays are

valuable for their ability to penetrate soft muscle tissue, but are obstructed by more dense

bone allowing for the use of x-ray images in the determination of bone fractures. More

important than the definition of the spectrum is the question, what makes up light?

Light consists of packets of energy called photons. These packets of energy

exhibit the utterly complex particle-wave duality, lending these particles to create

electromagnetic waves. Lights characteristics are determined by a set of unique values

belonging to each photon which render the photon properties of wavelength, frequency,

polarization, and energy. The relationship between a photons wavelength, , and

frequency, , are governed by the relationship

=

(1)

where c is the speed of light, approximately 2.9979108

meters per second in a vacuum.

However when traveling through a material c is reduced by n, the index of refraction of

the material.


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The energy of each photon is related to the wavelength by the equation

=

(2)

where

is Plancks constant, 6.626

10

-34

Joule seconds. The energy and wavelength of

the light used to create an optical trap, as well as the index of refraction of the trapped

material, will be important in determining the characteristics of the optical trap.

2.2 Introduction-Lasers

Laser is an acronym for Light Amplification by Stimulation Emission of

Radiation. Lasers were first hypothesized by Albert Einstein in 1917 in his bookZur

Quantentheorie de Strahlung (On the Quantum Theory of Radiation) [5]. Forty-three

years later, on May 16, 1960, Theodore Maiman brought Einsteins vision to life.

Working in the Hughes Research Laboratories, Maiman exposed a silver-coated ruby rod

to a high-power flash lamp. This exposure generated a very narrowband frequency

response, confirming the presence of the first laser beam.

A laser is a unique light source capable of generating a high intensity, low

divergence, coherent beam of monochromatic light. It may be advantageous to first

define a few terms here such as intensity, divergence, coherent, and monochromatic.

Intensity, also referred to as flux, is the wattage per area. For example, a low power laser

may easily have a higher intensity than a high power light bulb, in its bright spot, due to

the miniscule spot size of the laser versus the spray of the bulb. Divergence is the

measure by which the spot size of the beam increases as a function of distance between

two points. Low divergent light can easily be demonstrated with a cheap laser pointer.

When pointed at a surface centimeters away the spot size may be one centimeter in


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diameter, when observed

flashlight is a good exam

having the same phase an

wavelength or covering a

note that laser light is uni

possible.

An essential char

mode of the beam or the

electromagnetic modes (

of astigmatic spots or rin

Figure 1-TEM Modes [

meters away the spot size will be relatively the

ple of a highly divergent source. Coherent light

d frequency. Lastly, monochromatic light is li

small band of the electromagnetic spectrum. It

que and given these characteristics makes optic

cteristic of a laser, paramount to creating an op

beam shape. Beam shape is described by transv

EM) and can range from a singular spot to exo

s. Figure 1, below, displays a number of com

]

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same size. A

is defined as

ht of a single

is important to

al trapping

tical trap, is the

ersal

ic combinations

on modes.


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For creating an optical trap and for most other applications a TEM0,0 mode is most

desirable because it has a single intensity maximum at its center. The intensity of a

TEM0,0 laser beam follows the Gaussian distribution as shown below in Equation 3

= (3)whereIo is the optical axis intensity, ris the radial distance, w is the width of the

distribution andI(r) is the intensity. Many other subtle reasons of why laser light is

required for trapping will be explained later.

2.3 Introduction-Optical Trap

2.3.1 Introduction-Optical Trap-What is an Optical Trap?

An optical trap is a device which utilizes a highly focused laser in order to apply

piconewton forces to micro and nano-scale dielectric objects so as to constrain them to

the equilibrium position of the trap. These devices typically consist of a laser, a

microscope objective, and a series of mirrors and lenses to tailor the beam to the desired

specifications of the given application. An optical trap is capable of manipulating objects

as large as ten micrometers and as small as one-tenth of a nanometer. The term optical

trap is synonymous with optical tweezer and single beam gradient laser trap.

2.3.2 Introduction-Optical Trap-History

In 1970, Arthur Ashkin, a scientist working at Bell Laboratories first observed the

manipulation of micrometer sized particles through the use of a highly focused laser

beam. Throughout the 1970s and into the 1980s the optical trap was advanced to the


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point of creating a three dimensionally stable trap [5]. In the late 1980s Ashkin

discovered that optical trapping could be applied to biology, where the tobacco mosaic

virus and E. Coli were two of first biological specimens to be trapped. These advances

caused distinct changes in the types of experiments biologists could conduct, but many of

these applications were not implemented until the twenty-first century.

In the meantime, Steven Chu, a former colleague of Ashkin at Bell Laboratories

was awarded the 1997 Nobel Prize in Physics for his discovery of the magneto-optical

trap [5]. This variation of the optical trap is capable of holding particles with a diameter

of one-tenth of a nanometer. At the time this was one hundred times smaller than the

smallest particles trapped by conventional optical trapping methods. This type of

apparatus led to a number of interesting discoveries including Bose-Einstein Condensates

for which the 2001 Nobel Prize in physics was awarded.

As the twenty-first century came, so did the vast number of advances and

applications of optical trapping in conjunction with biology. Just as microscopes allowed

for advances in biology by granting scientists the ability to see microscopic objects,

optical traps granted scientists the capability of interacting with these microscopic

objects. Experiments involving bonding strength, elasticity of DNA, and cell sorting are

just a few of the new and growing number of applications in this field [3]. Even more

recently, emphasis has been put on smaller and cheaper optical trapping apparatuses.

Some of the cheaper optical trapping apparatuses for undergraduate use can be

constructed for 6,500 to 10,000 dollars [1].


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2.3.3 Introduction-Optical Trap-Classical Characterization

In its simplest and most general form, the force exerted on an optically trapped

particle can be characterized through Hookes Law, due to its restorative nature. When a

trapped particle is displaced from the center of the trap a force is exerted on the particle

which increases linearly in order to restore the particle to the center of the trap [8]. This

is analogous to that same particle being attached to a spring.

As a theoretically analogous example, an optical trap capable of exerting a five

piconewton force will be used, a common usable trap strength. For reference, the

apparatus that was constructed achieved a maximum trapping force of approximately

3pN. According to Hookes Law, as shown below in Equation 4, the force, F, is

proportional to the negative of the product of the spring constant, k, and the displacement,

x.

= (4)

Given the five piconewton force exerted and a maximum displacement of two micron,

typical in an optical trap, the theoretical spring constant is as follows in Equation 5.

=

= 0.025

(5)

The analogy of Hookes Law will sufficiently calculate the force exerted given a

specified displacement, but it does not explain the physics behind why the particle is

being trapped. Other more accurate and useful theoretical descriptions are advantageous

to explain in order to understand how and why an optical trap works.

There are three regimes that are useful to characterizing and describing how the

physics behind an optical trap work. The regime considerations are dependent on the


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ratio of the diameter of the trapped particle to the wavelength of light utilized to trap, the

Z ratio. The Rayleigh Regime is when Z > 1. The following sections will describe the

physics involved in each regime and provide examples of the possible strengths exerted.

2.3.4 Introduction-Optical Trap-Rayleigh Regime

The Rayleigh Regime, where the particle size is much less than the wavelength of

light, is best explained through electromagnetic theory. In this consideration the trapped

particle is treated as an induced dipole. The forces exerted on the induced dipole are the

gradient and scattering forces. The gradient force is expressed in the direction of the

intensity of the gradient while the scattering force is in the direction of beam propagation.

In order to achieve trapping, the gradient force must be greater than the scattering force.

The formula for the gradient force in the Rayleigh regime will be derived below.

The Lorentz Force Law, Equation 6, is the source of the gradient force.

= ( + ) (6)where q is the charge of the particle,Eis the electric field, v is the velocity of the particle,

andB is the magnetic field. With consideration to one charged particle Equation 6

becomes Equation 7.

= ( + ) (7)Equation 8 gives the polarization of a dipole wherep is the polarization and dis the

distance between the two charged particles.

= (8)


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Applying the dipole to Equation 7 gives rise to Equation 9.

= ( + () ) (9)

Since x1-x2 is very close to zero, the Taylor Series Expansion is taken, giving Equation

10.

= ( + + () ) (10)With the cancellation of E1 and E2 and application ofp the simplification arrives at

Equation 11.

= +

(11)Assuming that the dielectric material is linear it is helpful substitute Equation 12, where

is the linear dielectric constant, into Equation 11 to get Equation 13.

= (12) = +

(13)

Exploiting the substitution of the vector analysis equality in Equation 14 and the

Maxwell-Faraday, Equation 15, the solution is nearly in sight in Equation 16.

=

( ) (14)

=

(15)

=

+

( ) (16)Finally, Equation 17 provides the solution for the gradient force of the Rayleigh Regime.

=

(17)


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The solution dem

Rayleigh Regime, is subj

magnitude of the electric

It is the gradient of the b

In order to more f

gradient of a scalar field i

of increase of the scalar f

closer look at the applica

Figure 2, below, displays

analogous to a TEM0,0 be

Figure 2-Two Dimensio

When the particle is off c

center will draw the parti

onstrates that the gradient force on the particle,

ect to the dielectric constant of the object mater

field generated by the laser beam, and the gradi

am that gives rise to the restorative nature of th

ully understand this concept it is helpful to defi

s a vector field which points in the direction of

ield and whose magnitude is the greatest rate of

ion of a TEM0,0 beam gradient to Equation 17 c

a two dimensional Gaussian distribution of the

am.

nal Gaussian Distribution

enter, the increase in intensity profile of the bea

cle back to the center of the trap.

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when in the

al, the

ent of the beam.

e gradient force.

e a gradient. A

the greatest rate

change. A

onfirms this.

beam intensity

m towards the


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This is proven by taking the derivative of Equation 3, as shown below in Equation 18.

= 2( ) (18)The resulting plot of Equation 18, Figure 3 below, displays the magnitude and direction

of the force resulting from the gradient of a TEM0,0 beam.

Figure 3-Magnitude and Direction of Force on Trapped Particle vs. Displacement

Since the trap is symmetrical for a TEM0,0 laser beam mode, only one axis needs to be

differentiated in order to determine the force on the particle given a particular

displacement from the equilibrium position. Equation 17 is also useful in determining the

force on the particle when alternate gradient profiles in the beam are used. Alternative

gradient choices may include multiple TEM0,0 spots to trap multiple particles or even

time varying gradients to spin or rotate the particles in the beam.

In consideration of the scattering force, for most cases it is neglected. However, it

should be noted that in the case that the scattering force is greater than the gradient force,

the optical trap will become a particle accelerator [2]. Instead of holding the particle still,

the apparatus will push the particle in the direction of propagation of the beam. In most


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stable trapping cases the scattering force will cause the trapped particle to be displaced

slightly down the beam path such that the equilibrium position of the trap will be slightly

down from the beam waist.

2.3.5 Introduction-Optical Trap-Complex Regime

The complex regime is applicable when Z is equal, or very close, to one. Just as

the name implies, the Complex Regime is complex. The determination of how trapping

occurs and the theoretical force is resolved through the consideration of all of the lasers

electromagnetic field components, incident, scattered, and internal. Maxwells equations

are involved in these calculations and are far too complicated for discussion in this paper.

2.3.6 Introduction-Optical Trap-Mie/Ray Optics Regime

The Mie or Ray Optics Regime is applicable when Z is greater than one. This

trapping regime is the theoretical basis for the apparatus that was built for this

experiment. Ray optics utilizes changes in momentum of the incident beam due to the

difference in index of refraction of the sphere and surrounding material to determine the

force on the trapped particle as a result of Newtons third law.


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For the purpose o

780nm wavelength, simil

simplest ray optics case t

in water as shown in Fig

Figure 4-Simplest Ray

The index of refraction o

Equation 19, can be used

The transmission coeffici

In order to evaluate the c

must be calculated by fir

a theoretical example, specifications of a 40m

ar to that of the constructed apparatus, will be u

he beam is normally incident to a polystyrene s

re 4.

ptics Regime Diagram

water is 1.33 and the sphere is 1.55. Fresnels

to calculate the reflection and transmission coe

0.005835ent is as follows in Equation 20.

1 0.994165ange in momentum, the total momentum of th

t determining the momentum value of each pho

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beam and

sed. In the

here suspended

equation,

ficients.

(19)

(20)

incident beam

ton in the beam,


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then by considering the total number of photons in the beam per second. The momentum

of each photon is given by the DeBroglie wave equation and calculated as shown in

Equation 21.

=

= 8.495 28 (21)Equation 3 in conjunction with the definition of a watt as a joule per second allows the

calculation of the energy of each photon in Equation 22.

= 2.5467 19

(22)

Knowing that a 40mW beam is incident, dimensional analysis is used to calculate the

number of photons per second in Equation 23.

.

.

= 1.570717

(23)

Therefore the total momentum of the beam is approximately 1.3343E-10 N*s. Given the

reflection coefficient from Equation 19 the change in momentum is calculated in

Equation 24.

= 0.005835 1.3343 10 = 7.7856 13 (24)Given that this change in momentum is on a per second basis the seconds unit is divided

out to leave newtons. When converted into piconewtons, the change in momentum

results in approximately 0.7786 pN of force exerted on the particle from the normally

incident beam.

Now that it has been demonstrated that ray optics and change in momentum can

give rise to exerting a force on a dielectric sphere suspended in water it is useful to


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demonstrate how restorat

beam onto a sphere as sh

Figure 5-Off Axis Ray

Only regarding the first r

it is possible to determin

specifications as in the la

1.3343E-10 N*s in the +

incident reflections, the r

0.99084. Considering Sn

45, the first reflection ca

refractions result in a tota

ive forces can be generated. Consider a non-no

wn in Figure 5.

ptics Regime Diagram [6]

flected ray and the first and second refracted, t

the magnitude and direction of the force. Usin

st theoretical example, the total momentum of t

direction. Due to Fresnels Equation for non-

flection and transmission coefficients are R =

ells law and the incident angle of the beam ont

uses a 45 deflection in the X direction. The f

l beam deflection of 1.65 in the +X direction f

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rmally incident

ansmitted rays,

g the same beam

e beam is

ormally

.00916 and T =

o the sphere of

irst and second

om the incident


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beam. The next step is to determine the change in momentum for each beam and their

components. Equation 25, below, calculates the momentum of the reflected beam.

= 1.3343 10 0.00916 = 1.2222 12 (25)Since the transmitted beam contains X and Z components, sine and cosine functions must

be utilized. Equation 26, below, calculates the Z component of the transmitted beam.

= 0.99084 cos1.65 1.3343 10 =1.3215 10 (26)

Equation 27, below, calculates the X component of the transmitted beam.

= 0.99084 sin1.65 1.3343 10 =3.8068 12 (27)

The change in momentum for each component of the beam is calculated in Equation 28.

= 1.3343 10 1.32155 10 + 3.8068 12 1.222

12

= 1.275

12

+ 2.5848

12

(28)

This gives rise to forces of 1.275pN in the Z direction and 2.5848pN in the X direction.

When combined, the forces cause a net force of 2.8822 pN of the trapped particle in the

direction of 26.26 degrees vertically from the +X axis. Therefore the particle will be

restored towards the equilibrium position of the optical trap.


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A more comprehe

intensity profile is shown

Figure 6-Ray Optics Re

It is important to note tha

the top of the figure as w

In Figure 6 (a), the trap i

equilibrium due to the int

because of this important

drawn into the trap equili

There are a numb

previously discussed ray

First, the momentum cha

power. Therefore, an inc

nsive visualization with consideration of a TE

in Figure 6.

gime: Resulting Forces

t in Figure 6 the propagation of the beam is fro

ll as a beam intensity profile matching that of

characterized by a restorative force towards th

ensity profile of the TEM0,0 beam. F2 is much s

property. Figure 6 (b) exhibits how the trappe

brium when displaced along the beam path.

r of important conclusions that can be derived

optics theory that will be applied to apparatus c

nge of the incident beam is directly proportional

rease in beam power, Pbeam will result in an incr

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0,0 laser

the bottom to

TEM0,0 beam.

trap

ronger than F1

particle is

rom the

nstruction.

to the beam

ease in trap


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strength, Ftrap. Second, an increase in rate of change of the intensity of the beam, which

is accomplished through decreasing the diameter of the spot size, wtrap, will result in

increased trap strength. Equations 29 and 30 display these paramount relationships that

guide the construction of the apparatus.

(29) (30)


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3.0 Apparatus

3.1 Apparatus-Design

3.1.1 Apparatus-Design-Trapping Optics

In order to create the strongest optical trap possible, for the given components, it

is desirable to; first, maximize the power to the objective, and second, increase the

gradient of the intensity by decreasing the spot size [7]. Maximizing the beam power to

the objective is accomplished through shaping of the beam with lenses and minimizing

the subsequent loss. Choosing a laser with an output beam of TEM0,0 or as close as

possible to this mode will insure that the number of loss causing optical correcting

elements is minimized. On the other hand, increasing the gradient of the intensity is a

more difficult process.

The gradient of the intensity should be increased, but only to a certain point at the

expense of decreasing beam power. The method for increasing the gradient of the

intensity is through decreasing the spot size. However, there are a number of physical

limitations on the theoretically minimization of the spot size. These limiting parameters

are: NA, numerical aperture of the objective lens, n, index of refraction of the medium,

and the wavelength of light, o. Equation 31, below, calculates the theoretically

optimized spot size for a 1.25 NA microscope objective, index of refraction of 1.5, and

wavelength of 780 nm [7].

.

1 = 0.4208 (31)In order to achieve the minimized spot size, the diameter of the spot size, d spot,

must be equal to the diameter of the objective, Dobjective, at the position of the objective.


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In the case that Dobjective >

value. In the case that D

not reaching the objectiv

intensity of the beam will

match the diameter of the

element is commonly use

particular distance, as sh

collimated beam reachin

The final factor f

of the beam to that of the

when the radius of curvat

focal length of 16 cm pla

objective and result in th

producing this radius of c

wavelength of light 16 c

illustrates the basic optic

Figure 7-Basic Trappin

dspot, the objective will not focus the beam to t

bjective < dspot, significant loss may occur due to t

. This instance will form the minimized spot si

be decreased, lowering the strength of the trap.

spot size to the diameter of the objective a bea

d. The beam expander consists of two lenses s

wn below in Figure 7, in order to achieve an ex

to the objective.

r trap strength optimization is matching the rad

objective. For most microscopes, minimized s

ure of incident light is approximately 16 cm [7]

ed 32 cm from the objective will create a focus

proper radius of curvature. An alternative met

urvature is to place a pinhole with diameter of

from the objective [7]. For reference, Figure

l elements of an optical trap.

g Optics

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e minimized

he entire beam

ze, however the

In an effort to

expanding

parated by a

panded

ius of curvature

ot size occurs

. A lens with a

16cm from the

hod for

0 times the

, below,


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3.1.2 Apparatus-

Observation of th

well as for the collection

splitter is implemented al

ratios in the beam splitter

a 50/50 beam splitter will

image. A condenser or b

image for a charge-coupl

Most commonly

filter should be used to p

wavelength of light of th

CCD camera will filter o

spectrum for observation

system.

Figure 8-Basic Imaging

esign-Imaging System

optical trapping system is required for sample

of data. In order to observe the sample while tr

ong the beam path. Varying ratios of transmiss

may be used for particular applications, howev

allow for ample transmission while preserving

acklight source is used to illuminate the sample

d device (CCD) camera.

sed CCD cameras are robust enough for the ap

event damage to the CCD sensor. Depending

laser, a shortpass or longpass filter placed dire

t the majority of the laser beam while preservi

. For reference, Figure 8, below, demonstrates

System

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manipulation as

apping, a beam

ion to reflection

er for most cases

the reflected

and create the

lication, but a

n the

tly before the

g the visible

basic imaging


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3.1.3 Apparatus-Design-Sample Manipulation

If the apparatus exploits a complete microscope, coarse manipulation of the

sample is accomplished through the vertical adjustment of the objective and horizontal

manipulation of the sample stage. For a more precise solution and for apparatuses that do

not involve a complete microscope, a 3-axis XYZ translational stage is used to mount and

manipulate the sample. In order to achieve consistent stage manipulation speeds an

actuator with adjustable controller can be used. This will be necessary in testing the trap

strength later.

3.2 Apparatus-Utilized Components

3.2.1 Apparatus-Utilized Components-Laser

For the optical trap constructed in this experiment a 780 nm, 120 mW diode laser

was selected. A number of factors contributed to this decision including cost, power, and

wavelength. Diode lasers are very cheap and as an undergraduate thesis, cost cutting was

at a premium. The selected laser has ample power to achieve trapping while acquiring a

significant amount of loss due to the use of a number of optical elements. It is commonly

noted that 5 mW of power at the microscope objective is the minimum amount required

for trapping, while 20 mW will create a trap strong enough for many applications [1].

The 780 nm wavelength of the laser was advantageous to use because it was still within

the visible spectrum, allowing easy steering of the beam. Additionally, 780 nm is at the

edge of the visible spectrum permitting trouble-free filtering of the beam at the CCD

camera for imaging.


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The major disadvantage to using a diode laser is the production of poor beam

quality. The output beam was of the mode TEM0,1 as well as being astigmatic, a common

fault of diode lasers. Thus a number of lenses were implemented in order to optimize the

gradient of the intensity while still preserving beam power.

3.2.2 Apparatus-Utilized Components-Lenses and Mirrors

The two main lens sections are a beam expander to match the beam spot size to

the diameter of the objective and a 15 cm condenser lens with pinhole in order to match

the radius of curvature of the objective lens. A 10 cm and a 15 cm condenser lens were

placed 25 cm apart along the beam path to expand the beam to approximately .75 cm, the

diameter of the microscope objective. A number of mirrors were used to steer the beam

into a vertical orientation such that the 15 cm condenser lens, pinhole, and microscope

objective were all in the same plane. A vertical adjustability of 5cm was implemented

into the mounting of the 15 cm lens to allow for the changing of the spot size and radius

of curvature at the objective. 3-axis adjustability was employed in the mounting of the

pinhole to insure optimum alignment with the focus of the beam from the 15 cm

condenser lens. All lenses and mirrors used were AR coated to reduce loss.

3.2.3 Apparatus-Utilized Components-Objective Lens

The microscope objective used was a 100x, 1.25 NA objective. The diameter of

the objective was 0.75cm. The radius of curvature of the lens was 16cm.


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3.2.4 Apparatus-Utilized Components-Imaging System

A CCD camera was used to capture the image, while a 50/50 beam splitter was

used to send the image into the CCD camera, and sample illumination was carried out

through a backlight source. The image was filtered to preserve the image and protect the

CCD sensor with a 750nm shortpass filter. The CCD camera was interfaced to a monitor

as well as to a USB MPEG Frame Grabber for connection to a computer. The converted

video was recorded with a Software Development Kit provided by the manufacturer.

3.2.5 Apparatus-Utilized Components-Sample Manipulation

Sample manipulation was made possible through the use of a 3-axis, XYZ

translational stage. Precise manipulation was conducted with a motorized actuator wired

to a jog/run actuator controller. This component is important for accurate sample

manipulation and ultimately trap strength measurement.


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When completed the trapping, imaging, and manipulation portions of the constructed

apparatus are all as shown in Figure 9.

Figure 9-Optical Trap Diagram


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4.0 Experiment

4.1 Experiment-Sample Preparation

The following tests were conducted with three-micron diameter polystyrene

spheres in water solution. The microspheres were selected for their relative low cost,

ease of trapping, and availability of reference numbers for data analysis. Since the

spheres came in a dense solution, a diluted mixture was produced by mixing the

microsphere solution and deionized water at a ratio of 1:40. This allowed for enough

microspheres to reside, for reference, in the visual field of the CCD camera, while

leaving plenty of room for unobstructed translation of the trapped microsphere.

One drop of the diluted solution, approximately 50 microliters, was placed on the

slide and covered with a slip cover. The microspheres will sink in solution and trap

strength decreases as the depth of the trap increases. Therefore, no more than 50

microliters of sample solution should be used to avoid creating a slide that is too deep for

trapping. Index matching fluid was placed on top of the slip cover. Figure 10 displays

an image of the microspheres in the undiluted solution.

Figure 10-Image of Undiluted Microspheres in Solution


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5.0 Data and Ana

5.1 Data and Ana

As previously stat

USB MPEG Frame Grab

translated and fell out of

analyzing the timestamp

had travelled over that ti

image. The known size

distance travelled. Figur

particle is determined.

Figure 11-Determinatio

ysis

ysis-Force Calculations

ed, the tests were recorded through the use of a

ber, and a computer. As the trapped microsphe

he trap, the translational velocity could be dete

of each frame as well as knowing how far the tr

e span by referencing stationary microspheres

f the microspheres, three micron, was helpful i

11, below, shows two images and how the vel

n of Microsphere Velocity by Image Analysis

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CCD camera, a

es were

mined through

apped particle

within the

determining the

city of the


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The average diameter of the microspheres was found to be 21 pixels. By referencing the

microsphere diameter in pixels, a scale of 7 pixels per micron was determined. Finding

the distances travelled was accomplished by noting the pixel location on the rightmost

point of the microsphere from the first image and drawing a line from that point in the

second image to the rightmost pixel of the same microsphere in the second image. The

difference between these two values provides the distance travelled in number of pixels.

Equations 32, 33, and 34 demonstrate how given the frame stamps and number of

translated pixels the average velocity of the particle can be determined.

=

= 18.857 (32)

=

= 0.333 (33)

.

.= 56.628

(34)

Given the velocity of the particle, Stokes drag force, Equation 35, is used to

calculate the drag force exerted on the trapped microsphere.

= 6 (35)Where is the drag force, is the viscosity of water, 10-3 N*s/m2, is the radius ofthe microsphere, and is the velocity of the microsphere [7]. For the given values andthe velocity of the particle from Figure 10 at 56.628 micron per second the drag force is

approximately 1.699 pN. The trap strength is determined by the maximum velocity of

the particle, and is equal to the drag force at that maximum velocity.


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5.2 Data and Analysis-Data

The position of the 15 cm lens was held constant while the size of the pinhole was

altered. This was tested in order to determine whether increases in the quality of the laser

beam mode would result in increases in trap strength versus increased power at the

objective. The results from section 4.2 Experiment-Pinhole, are as shown below in

Figure 12.

Figure 12-Pinhole Experiment Results

While only two data points appear on the graph, a third which represents the test

with no pinhole is represented as a number much greater than 100 micron. As a point of

reference for the no pinhole data point, the trapping strength calculated was 2.245 pN.

The best fit line was determined from a TEM0,0 laser beam mode. This was given the

assumption that as a perfect beam mode, the trapping strength would be at a maximum

when no pinhole was placed in the apparatus, resulting in maximum power at the

0

0.5

1

1.5

2

2.5

3

0 50 100 150 200 250 300 350 400 450 500

Force(pN)

Pinhole Size (micron)

Trap Strength vs. Pinhole Size

No pinhole


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It was determined that the maximum trapping strength achieved was 3.1 pN. It is curious

to note the asymmetrical nature of the graph. It is possible that this is a result of an

overexpansion of the diameter of the beam in the beam expander. This is because as the

15 cm lens position came closer to the objective the trap strength increased, possibly

indicating a more direct match in the size of the diameter of the beam spot size and the

diameter of the objective. As the 15 cm lens position was moved further from the

objective, overfilling of the objective leads to a loss of power incident on the objective

lens, resulting in the loss of trapping strength.


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6.0 Conclusions

An optical trapping apparatus was successfully constructed in order to conduct the

experiments. With this apparatus, the experiments were conducted and it was determined

that a maximum trapping strength of 3.1 pN was exerted by the trap. The resulting trap

strength is reasonable considering the poor beam mode of the diode laser and the number

of optical elements required to optimize the beam for trapping. In comparison to a

known commercial apparatus from Thorlabs which achieves approximately one

piconewton of trapping strength for every 6.72 mW of power incident at the objective,

the constructed apparatus at maximum strength maintains 8.87 mW of power at the

objective for one piconewton of trapping strength. The constructed apparatus requires

approximately 32% more power per piconewton of trapping strength.

The increase in power per piconewton of force is attributed to the poor beam

quality. Even after clarifying the beam through a number of optical elements the beam

mode was hardly ideal. A better beam mode of TEM0,0 or much closer could be achieved

by replacing the diode laser, in the apparatus, with a high power Helium-Neon (HeNe)

laser which naturally produces a TEM0,0 beam. However, high power HeNe lasers are

very expensive and were unable to be used for this experiment due to budget constraints.

Given the budget constraints of the experiment, the construction of the optical

trap for such little money was, in itself, a success. Tabulating the cost of all the parts

used in the experiment, the total cost was approximately 5,000 dollars. However, a

number of the high cost components such as the microscope objective lens, motorized

actuator, and actuator controller were already available in the laboratory, decreasing the

expenses to construct the apparatus to less than 2,000 dollars. Compared to low cost


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apparatuses for undergraduate construction found in multiple pieces of literature, these

costs ranged from 6,500 to 10,000 dollars [1]. The Thorlabs optical trapping kit, in order

to have all of the same features as the apparatus constructed for this experiment, would

cost 19,200 dollars. While the Thorlabs apparatus is capable of exerting approximately

20 pN of force, the constructed apparatus achieved more than adequate strength for the

experiments conducted.


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7.0 Acknowledgements

First, I would like to thank my advisor, Dr. Todd B. Smith. His guidance by

assisting me in the gathering of journal articles, helping me in the laboratory, and in the

preparation of this thesis ensured its successful completion and more importantly, my

increased knowledge and experience gained from this process. I cannot thoroughly

express enough, my gratitude for his continued support, especially with the blessing of a

second child coming to him and his wife during this process. His passion for physics

truly inspired me during this process and has been a source of inspiration even during the

most frustrating times in the laboratory.

Secondly, I would like to thank a number of university departments and faculty

members. Thanks to the University of Dayton Honors Department for their monetary

support, allowing for my continued research during the summer as well as for the

purchasing of a number of devices used in the construction of the apparatus. I would like

to thank the Physics Department, especially Dr. Rex Berney, Dr. Bob Brecha, Dr.

Yiqiong Zhao, and Dr. Perry Yaney. The additional funding made available by Dr.

Berney and the generous support of the department allowed the full realization of the

construction and testing of this apparatus to occur. To Dr. Brecha and Dr. Zhao, thank

you for the use of laboratory space. And to Dr. Yaney, thank you for the extra guidance

and assistance in finding many of the parts in and around the laboratory.

Lastly, I cannot forget to acknowledge the support of my fellow students, friends,

roommates, and most importantly family members who have dealt with absence at times

during this process. And to anyone reading this, thanks for taking interest. I hope you

now know a little more than you did before.


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8.0 Works Cited

1. Bechhoefer, John, and Scott Wilson. "Faster, Cheaper, Safer Optical Tweezers for the

Undergraduate Laboratory."American Journal of Physics 70.4 (2002): 393-400.

Print.

2. Cotterill, Rodney. "Chapter 6: Some Techniques and Methods."Biophysics an

Introduction. Baffins Lane: Wiley, 2002. Print.

3. Greulich, K. O.Micromanipulation by Light in Biology and Medicine: the Laser

Microbeam and Optical Tweezers. Basel: Birkhuser, 1999. Print.

4. Laguerre-Gaussian. Digital image. 8 July 2007. Web. 3 Dec. 2009.

5. McGloin, David. "Optical Tweezers: 20 Years on." Philosophical Transactions of the

Royal Society (2006): 3521-537.Royal Society Publishing. Web. 23 Mar. 2009.

6. Optical Trap Ray Optics Explanation. Digital image.

Http://en.wikipedia.org/wiki/Optical_tweezers. Web. 3 Dec. 2009.

7. Smith, Stephen P., Sameer R. Bhalotra, Anne L. Brody, Benjamin L. Brown, Edward

K. Boyda, and Mara Prentiss. "Inexpensive Optical Tweezers for Undergraduate

Laboratories."American Journal of Physics 67.1 (1999): 26-35. Print.

8. Zhang, Hu, and Kuo-Kang Liu. "Optical Tweezers for Single Cell."Journal of the

Royal Society (2008): 671-90.Royal Society Publishing. Web. 23 Mar. 2009.

kevin george thesis final draft

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