modelling a polystyrene bead trapped with a laser diode as a simple harmonic oscillator

8/13/2019 Modelling a polystyrene bead trapped with a laser diode as a simple harmonic oscillator

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Modelling a polystyrene bead trapped

with a laser diode as a simple harmonic

oscillatorGoh Wei Zhong

5 May 2013

Lab partner: Yizhou Shen

AbstractA focused laser beam generated by a laser diode was used to trap polystyrene

beads of diameter 1.7 m suspended in heavy water. This trap was modelledas a simple harmonic oscillator, and the regression functions for the effective

spring constant and the maximum trapping force were found; both were found

to increase with laser intensity. The trap radius was found to be around 0.5 m,

and beyond this the simple harmonic oscillator model must break down.


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Theory

Bead suspended in a fluid

Figure 1: A dielectric spherical bead, whose centre is indicated by a black dot, suspended in a fluid of a lower refractive index.

The solid arrows represent the incident rays and the exiting refracted rays, the dashed arrows represent rays travelling

within the bead, and the dashed lines represent the tangents to the bead at each bead-fluid interface. Adapted from Smith

et al (1999), Figure 1.

A dielectric is a poor conductor of electrical current, whose charges separate when an electric field is

applied in such a way that reduces the electric field within it.Figure 1 shows a dielectric spherical bead

of radius larger than the wavelength of visible light, suspended in a fluid of a lower refractive index.

When a ray of light r0, entering at an angle from the vertical, is first incident on the bead, a

component r1would be externally reflected and exit the bead, and the rest would be refracted. The

refracted ray would travel within the bead until it encounters a boundary with the fluid, when a

component r2would be refracted, and the rest would be internally reflected. The reflected ray travels

within the bead, and the process continues.

According to quantum theory, light travels as photons, which carry momentum. The total momentum of

the rays exiting the bead in various directions,ep

, is in general not the same as the momentum of the

incident ray,0

p

. From the principle of conservation of momentum, the bead encounters a change in

momentum,bead

p

, given by

eppp

0bead

.

r1

r3

r4

incident ray, r0

externally reflected ray,

first exiting refracted ray, r2

internal reflection


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Treating our system with such generality forces us to consider the momentum changes associated with

many rays, which is a complicated process. However, we may simplify our treatment by noting the

following:

With each internal reflection of light within the bead, the light intensity decreases, such that themomentum changes associated with the exiting refracted rays after the first becomes lesssignificant.

With increasing , the difference in polar angle of the externally reflected ray and the incidentray decreases, so the net momentum change associated with the externally reflected ray

decreases with .

On the other hand, with increasing , the difference in polar angle of the first refracted ray andthe ray within the sphere before the refraction increases, so the net momentum change

associated with the first exiting refracted ray increases with.

This analysis shows that for large enough , the effects of the

first exiting refracted ray eclipses that of the externallyreflected ray. Actually, this holds as long as the incident ray

has a sufficiently large perpendicular distance from the

centre of the bead. Otherwise, as shown inFigure 2,there

would hardly be any refraction as the light ray exits the bead:

there would be too little change in polar angle between the

first refracted ray and the ray within the sphere before the

refraction. This would mean that the momentum change due

to the externally reflected ray would become significant. We

will postpone this case to the middle of the following section.

For now, we will consider incident rays with a sufficientlylarge perpendicular distance from the centre of the bead,

such that to gain a qualitative understanding, we may neglect

the effects of the externally reflected ray, and need to only

consider the momentum changes associated with the first exiting refracted ray. In other words, we may

approximate the momentum change of the bead by

20bead ppp

, (large )

where2p

is the momentum of the first exiting refracted ray.

Optical tweezersIn a basic optical tweezers setup, a focused laser beam is incident on a dielectric spherical bead. The

momentum changes that the bead experiences may be found by considering a pair of rays incident on

the bead at an angle from the vertical in opposite directions.Figure 3 shows ray tracing diagrams of a

pair of rays incident on a bead when its centre is displaced by a sufficient distance to the (a) bottom, (b)

top and (c) left of the pointsf, the foci of the rays in the absence of the bead. We only consider the

momentum changes associated with the first exiting refracted ray.

Figure 2: An incident ray that has a small

perpendicular distance from the centre of the

bead hardly refracts as it exits the bead. In thiscase, the momentum change due to the

externally reflected ray would have to be

considered as well.


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Figure 3: A dielectric spherical bead suspended in a fluid of a lower refractive index. The thick solid arrows represent the

incident rays, the dashed arrows represent rays travelling within the bead, and the thin solid arrows represent the first

exiting refracted rays. The points markedfare the foci of the rays in the absence of the bead. Adapted from Ashkin (1992),

Figure 2.

InFigure 3a, from symmetry, the bead experiences no horizontal momentum change. On the other hand,

the projection onto the vertical axis of unit length of the reflected rays (thin solid arrow) is larger than

that of the ray travelling within the bead (dotted arrow). This means that the light rays pick up

downward momentum as they refract and exit the bead. From the principle of conservation of

momentum, the bead experiences an upward momentum change, towards the pointf.

InFigure 3b, the bead experiences no horizontal momentum change likewise. The projection onto the

vertical axis of unit length of the ray travelling within the bead (dotted arrow) is larger than the reflected

rays (thin solid arrow). This means that the light rays lose downward momentum as they refract and exit

the bead. From the principle of conservation of momentum, this downward momentum is transferred tothe bead, pushing it towards the pointf.

InFigure 3c, by considering the projection onto the vertical axis of the light rays as above, we find that

the ray incident from the left causes the bead to experience a downward momentum change, and the

ray incident from the right causes the bead to experience an upward momentum change, as they refract

and exit the bead. A careful calculation would reveal that these changes cancel exactly, so there is no

vertical momentum change. By considering the projection onto the horizontal axis of the light rays as

above, we find that both incident rays impart a leftward momentum change onto the bead toward the

pointfas they refract and exit the bead.

We now return to the case where the incident rays have a sufficiently small perpendicular distance from

the centre of the bead. In other words,fis sufficiently close to the centre of the bead. In this case, there

is a downward momentum change associated with the externally reflected rays, which travels upwards

and sideways, which becomes significant.

Synthesising the preceding discussion, we find that when the centre of the bead is at a point just belowf,

the downward momentum change associated with the externally refracted ray exactly cancels the

(b) (c)(a)


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upward momentum change associated with the first exiting refracted ray as depicted inFigure 3a. At

this position, the particle is in equilibrium. A small displacement of the centre of the bead from this

equilibrium position results in a momentum change, and hence a force, that restores the particle

towards equilibrium. We thus have a bead kept in stable equilibrium by a focused laser beam. We say

that the particle is in a trap.

Simple harmonic oscillator model for a trap

Since the bead in a trap is in stable equilibrium, then for sufficiently small displacements from the

equilibrium, we can always model the trap as a simple harmonic oscillator. The particle is free to move

in two directions, which we will consider separately. We thus have that the particle in a trap is subject to

restoring forces Fxhorizontally and Fyvertically given by

xkF xx and ykF yy ,

wherexand yare sufficiently small horizontal and vertical displacements from equilibrium and kxand ky

are the horizontal and vertical spring constants. The potential energy of the bead for the horizontal and

vertical oscillators are given by

2

2

1xkU xx and

2

2

1ykU yy .

Since potential energies each have one quadratic degree of freedom, the equipartition theorem states

that in thermal equilibrium,

TkykUxkU Byyxx2

1

2

1

2

1 22 , (1)

where kBis the Boltzmann constant and Tis the thermodynamic temperature of the system.

The ray optics of our system implies that there is a certain maximum restoring force Fmaxthat the laser

beam can apply on the bead. We may measure this force by subjecting our bead to a drag force given by

the equation for Stokes drag,

rvFD 6 , (2)

where is the dynamic viscosity of the medium, r is the radius and vthe velocity of the bead. The

maximum drag force that the bead can withstand while still remaining in the trap is then equal in

magnitude to Fmax. Within the simple harmonic oscillator model for our trap, we can then define the trap

radius rtrapsuch that

trapmax krF . (3)

For a displacement of the bead beyond rtrap, the laser beam would be unable to provide a restoring force

towards the equilibrium, of a magnitude that would be consistent with the simple harmonic oscillator

model.


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Apparatus

Figure 4: A schematic view of the experiment. Dotted lines represent data connections.

Figure 4 shows a schematic view of our experiment,Figure 5 shows a photograph of our setup, while

Table 1 summarises the roles of the various components shown:

Equipment Role

Laser generator Control intensity of laser output

Cover slide Reflect a fraction of laser power to intensity detector

Intensity detector Detect intensity of reflected portion of laser

Mirrors Reflect laser onto stage, tilt to change laser position on the stage in

response to joystick

Joystick Change the laser position on the stage so that bead can be trapped

Lenses Ensures that laser beam is focused on the stage

Signal generator Generate triangle waves for the moving trap part of our experiment,

described below

Digital oscilloscope Detect triangle wave frequency

Webcam Capture video of beads on the cover slide as magnified by the objective

Filter Filter out the red light so that the laser beam is hidden from the

captured video

Intensity

detector

WebcamFilter

Objective

Sample

Stage

Adjusting knobs

Lamp

Microscope

Base

Lens

Mirror

Laser output

Cover slide

Joystick

Digital

oscilloscopeSignal generator

Laser generator

To computer


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Objective Magnify the contents of the cover slide

Sample House a suspension of 1.7 m polystyrene beads in heavy water

Stage Serve as a platform for the cover slide

Adjusting knob Allow the objective to be moved relative to the stage so that different

parts can be captured by the webcam

Lamp Illuminate the cover slide so as to brighten the image captured by thewebcam

Base Serve as a stable mounting platform for the entire setupTable 1: Summary of roles of various components in our experiment.

Figure 5: Photograph of the setup for laser tweezers. The blue path is the main laser beam that traps polystyrene beads. The

red path is a beam that is reflected off a cover slide and travels to an intensity detector. Taken by Keith Zengel.


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Procedure

Stationary trap

With the laser power turned up on the laser generator, a 1.7m spherical polystyrene bead was

trapped and isolated. With the laser power adjusted such that 1.0 mW of power was reflected into the

intensity detector, a video of around 500 frames of the beads motion due to Brownian effects was

recorded using the webcam. The same process was repeated with 0.8 mW, 0.6 mW, 0.4 mW and 0.2

mW of reflected laser power respectively. An ImageJ macro was then used to determine and tabulate

the centres of the bead as captured in each frame.

Moving trap

With the laser power turned up, a bead was trapped and isolated. The laser power was adjusted to give

1.3 mW of reflected laser power. A signal generator was then turned on and set to deliver a triangle

wave of amplitude that resulted in a 1.2 m vertical movement of the trap position. The frequency of

the triangle wave was gradually increased from zero until the bead falls out of the trap, at which point

the triangle wave frequency was recorded. This process is repeated to give two more frequency readings.

The process above is repeated with 1.1 mW, 0.8 mW, 0.6 mW, 0.4 mW and 0.2 mW of reflected laser

power respectively.


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Results

Stationary trap: Effective spring constants

For a certain laser power, the distribution of particle positions across 500 frames approximates a normal

distribution, as shown inFigure 6.Figure 7 shows the variance in bead position as a function of laser

power; the larger the laser intensity, the smaller the variance in bead position. For the remainder of this

report, 1.0 unit of laser power is equivalent to the laser whose reflected part has a power of 1.0 mW.

From Eq(1) and taking T= 297 K, we can calculate the effective spring constant from the variance of the

bead position. For a laser with 1 mW of reflected power, we have a variance in the horizontal bead

position of 16107.6 m2and thus an effective horizontal spring constant of

1616

23

2mN101.6

107.6

2971038.1

x

Tkk

B

x.

Figure 6: The distribution of particle positions over 500 frames approximates a normal distribution. The distribution for a

laser with 0.2 mW of reflected power (blue) is wider than that for a laser with 1.0 mW of reflected power (purple).


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Figure 7: Variance in bead position as a function of laser power. The larger the laser intensity, the smaller the variance in

bead position.

The spring constants are shown inFigure 8 as a function of laser power. The more intense the laser is,

the higher the effective spring constant. Excel gives the linear regression functions for the spring

constants in the horizontal direction, kx, and in the vertical direction, ky, in mN m1, as

43.081.064.077.6 Pkx , (4)

9.16.39.28.16 Pky , (5)

where Pis the laser power in units.

The proportional relationship between laser intensity and effective spring constant is consistent with

theory. As the laser intensity increases, the number of photons generated by the laser beam per time

that traps the bead increases. Since each photon causes a fixed momentum change, the momentum

change per unit time (that is, the force on the bead) increases for a constant displacement of the bead

from the equilibrium. This corresponds to a larger spring constant.

The y-intercepts should be near zero as no trap would form without a laser. The fact that the y-

intercepts are two standard deviations away from zero is indicative of experimental error, to be

discussed later.

0

1

2

3

4

5

0 0.2 0.4 0.6 0.8 1 1.2

Varianceinbeadposition/1015m

2

Laser power/Units

x-axis y-axis


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Figure 8: Effective spring constant of trap as a function of laser power. The more intense the laser is, the higher the effective

spring constant.

The vertical spring constants are around double

that of the horizontal spring constants. This is

largely because the laser from a laser diode

diverges more rapidly in the vertical direction

than in the horizontal direction.Figure 9 shows

a photograph of the laser beam passing through

no objective, shining on a piece of white paper

on a completely lowered microscope stage.

Since our setup uses circular lenses, the laser

beam becomes elliptical and elongated in the

vertical direction, thus localising beads more

effectively in the vertical direction than in the

horizontal direction.

Maximum drag force

The maximum frequencies at which a trapped

bead remains trapped over three trials is

tabulated inTable 2,and the average of these

three trials is plotted inFigure 10.The more

intense the laser is, the higher the frequency a trapped bead can withstand.

kx= 6.77P- 0.806

ky = 16.8P- 3.63

0

2

4

6

8

10

12

14

16

0 0.2 0.4 0.6 0.8 1 1.2Effec

tivespringconstant/10

6N

m1

Laser power/Units

x-axis y-axis Linear (x-axis) Linear (y-axis)

Figure 9: A photo of the laser beam when allowed to diverge.

Laser from a laser diode diverges more rapidly in the vertical

direction than in the horizontal direction.


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Laser power/Arbitrary units 0.2 0.4 0.6 0.8 1.1 1.3

f1/Hz 8.0 17 30 33 48 58

f2/Hz 7.5 17 23 30 42 47

f3/Hz 7.3 16 22 36 43 50Table 2: Maximum frequencies of a triangle wave applied to the laser controller, at which a trapped bead remains trapped.

Figure 10: Average maximum triangle wave frequency as a function of laser power. The more intense the laser is, the higher

the frequency a trapped bead can withstand.

0

10

20

30

40

50

60

0 0.5 1 1.5

Maximumt

rianglewave

frequency/Hz

Laser power/Units


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The drag force can be calculated from Eq(2),using the value of dynamic viscosity of heavy water, 1.25

mPa s. For example, for the 1.1-unit laser, we have an average maximum triangle wave frequency of

44.3 Hz, and so

N.107.73.44102.12105.800125.066 1267 rvFD

The maximum drag force encountered by a bead within the trap is plotted as a function of laser power

inFigure 11.The more intense the laser is, the larger the drag force a trapped bead can withstand. Excel

gives the linear regression function of the maximum drag force, Fmax, in piconewtons, as

092.0074.011.092.5max

PF , (6)

where Pis the laser power in units.

Trap width

The trap width can be calculated from the regression functions for kygiven in Eq(5),and for Fmax, given

in Eq(6).For instance, for the 1.1-unit laser, we have a trap width of

m1045.0

106.31.18.16

10074.01.192.5 6

6

12

max

F .

Figure 11: Maximum drag force encountered by a bead within the trap as a function of laser power. The more intense the

laser is, the larger the drag force a trapped bead can withstand.

Fmax= 5.92P+ 0.074

0

1

2

3

4

5

6

7

8

9

0 0.5 1 1.5

Maximumd

ragforcewithin

trap/1012N

Laser power/Units


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For the 0.2-unit laser, the regression function for kygives a negative value so the measured value at 0.2

units, 1.3 mN m1, was used instead:

m1097.0

103.1

10074.02.092.5 66

12

max

F .

The trap width is plotted as a function of laser power is plotted inFigure 12.From the data for laser

power of 0.6 units and above, we see that the trap width is around 0.5m, or around 1/3 the size of the

polystyrene beads, consistent with literature. In other words, if we model the trap as a simple harmonic

oscillator, a 0.5 m displacement would result in the maximum restoring force on the bead, and any

larger displacement would allow the bead to fall out of the trap.

For laser powers of 0.2 and 0.4 units, the trap width is calculated to be around 1.0 m and 0.8 m

respectively, which is larger than the values for more intense lasers. This is likely a result of experimental

error leading to the negative y-intercept in the regression function for ky, increasing Fmaxsignificantly for

smaller values of laser power. The value of Fmaxwould have been roughly constant if the regression

function for kypassed close to the origin, as theory predicts.

Figure 12: Trap width as a function of laser power.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5

Trapwidth/m

Laser power/Arbitrary units


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Sources of error

One significant source of error is the large variation in the laser power as the experiment was conducted.

For laser power of 0.8 units and above, the power meter fluctuated by as much as 0.1 mW over seconds.

It was thus difficult to determine the mean laser power. This error could have been reduced by

repeating the stationary trap experiments several times for large values of laser power, with the laser

power reset and adjusted each time. This could have resulted in a better fit for the effective spring

constants inFigure 8.Indeed, we repeated the moving trap experiment several times, averaged our

results over those trials and obtained a better fit for the maximum trap force inFigure 11.

ReferencesAshkin, A. (1992). Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics

regime. Biophys. J., 61(2), 569-582.

Smith, S. P., Bhalotra, S. R., Brody, A. L, Brown, B. L., Boyda, E. K, & Prentiss, M. (1999). Inexpensive

optical tweezers for undergraduate laboratories.Am. J. Phys. 67(1). 26-35.

modelling a polystyrene bead trapped with a laser diode as a simple harmonic oscillator

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