Final Project Report

Calibration of an Optical Tweezer Apparatus
A. Stange
(260265973)
R. Muller-Moran
(260321571)
Supervisor: Prof. Walter Reisner
April 16th
, 2012
Abstract
A 1064 nm Class IV CO2 laser was used in an optical tweezer set up. The trap stiffness
was calibrated by measuring the time series of a trapped polystyrene microbead and
analyzing its corresponding power spectral density function. Calibration measurements
included determining the effects of surface proximity and confinement time on trap
stiffness, ktr. Preliminary measurements of the trap stiffness were found to be ktr =
0.00591 ± 0.00139 pN/nm in the x direction and ktr = 0.00693 ± 0.00068 pN/nm
in the y direction. However, the accuracy of these results is still questionable due
to certain issues with the apparatus which remain to be resolved. Additionally, the
effect of surface proximity on backscattered laser light was investigated. Once properly
calibrated, the apparatus can be used to measure forces on the piconewton scale.
i

Contents
1 Introduction 1
2 Theory 2
2.1 Optical Tweezers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.1 Gradient Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1.2 Scattering Force . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Force Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 Experimental Methods 7
3.1 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.1 Quadrant Photodiode (QPD) . . . . . . . . . . . . . . . . . . . . 7
3.2 Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
4 Results and Discussion 11
4.1 Beam Proﬁle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Corner Frequency vs. Bead Proximity to an Edge . . . . . . . . . . . . . 13
4.3 Corner Frequency as a Function of Time . . . . . . . . . . . . . . . . . . 14
4.4 Backscattered Light Intensity vs. Bead Proximity to Edge . . . . . . . . 16
5 Conclusion 18
6 Acknowledgements 18
7 Author Contributions 18
A Stoke’s Law 20
B Percent Diﬀerences Between x and y Corner Frequencies 21
C Backscattered Light Intensity vs. Bead Proximity to Edge 23
ii

1 Introduction
Optical tweezers (OT) involve focusing a powerful laser beam in order to trap and ma-
nipulate objects on the micro and nano scale. The first optical tweezer apparatus was
achieved in 1970 by Arthur Ashkin [1]. The technology now provides a widely used
method to trap and move objects in three dimensions if they are sufficiently small.
Additionally, if the OT is calibrated, it can be used to measure minute forces on the
objects which it is currently trapping.
Measuring the force exerted on a bead as a function of the beam location would pro-
vide a simple way of mapping out the topography of a small three-dimensional volume.
This technique is called Photonic Force Microscopy (PFM)[2]. Although this is not a
new method, Professor Reisner is interested in using PFM to image single molecules
in a nanofluidic device (which one cannot do with other imaging techniques such as
Atomic Force Microscopy).
Professor Reisner also plans on using the OT for “Nanodozer Experiments” which
involve using a trapped bead to adjust the size of a nanocavity confining single or mul-
tiple DNA molecules. For example, a property of DNA which has never been measured
before is its force-compression curve—the force that DNA exerts as a function of its
compression. This could be measured by confining a strand of DNA in a nanochannel,
driving it against the back of the channel with a trapped bead, and then measuring
the force exerted on the bead. This is of interest because DNA exists in biological
systems in very high concentrations in the nuclei of cells. By creating an artificial
confinement environment for DNA where parameters such as cavity size and DNA size
can be adjusted, the physics of confined DNA can be better understood (see Figure 1).
FDNA
adjustable channel length
Figure 1: Depiction of a Nanodozer Experiment involving confined DNA in a nanochannel. The
trapped bead can be used to adjust the confinement volume by varying the channel length. The
force exerted by the confined DNA on the bead, FDNA, can also be measured.
1

A measurement of the force-compression curve of DNA would also provide insight
into the entropic behavior of confined DNA—specifically, the mechanism by which
multiple DNA molecules in confinement segregate during cell division. By running
Nanodozer experiments with multiple confined polymers, the phase-space of polymer
mixing/de-mixing could be determined as a function of confinement.
Finally, an interesting property of DNA is that it forms knots upon confinement (just
like string in your pocket). Professor Reisner also plans to study the formation of these
knots in adjustable nanocavities.
Because of the sensitivity involved in these types of experiments, it is important to
ensure that the optical trap is properly calibrated for force measurements. Depending
on the nature of the desired experiments, different factors may affect the calibration of
the device. For instance, nanochannel and PFM experiments involve close proximity to
a wall or surface. Thus, the main focus of this project was to determine how different
environmental effects could influence the calibration of the trap.
2 Theory
2.1 Optical Tweezers
Fundamentally, optical trapping occurs when the electromagnetic field of a focused
laser is able to induce dipoles in a dielectric material. Quantum mechanical considera-
tions are needed in order to fully explain this phenomenon. However, optical tweezers
can also be understood from the principle of conservation of momentum; laser beam
photons experience refraction at the surface of trapped objects, resulting in photon
momentum changes which are then imparted to the trapped objects themselves. Force
potentials then form as a result of momentum changes in different directions. Cylindri-
cal coordinates, taken with respect to the direction of propagation of the laser beam,
can be used to clarify the momentum changes which occur during trapping. Thus, pρ
denotes radial momentum and pz denotes axial momentum.
In practice, a component of the laser beam does not refract off of the trapped particle
but is instead reflected. This backscattered light is important in trapping experiments
because it is used to detect the position of the trapped object (see Section 3.1), but
plays no role in trapping theory and is thus ignored in this section. Keep in mind,
these forces are extremely weak and can only trap sufficiently small objects.
2.1.1 Gradient Force
Consider a small microbead at rest in an aqueous solution (in reality, such a bead would
be moving with Brownian motion but we can approximate it as being stationary). In
order to attempt trapping the bead using an unfocused laser beam, we need to consider
the intial momentum of the bead and photons. Since the bead is stationary and the
laser beams are travelling in the axial direction only, the net momentum of the system
is p(ρ, z) = (0, pz).
2

r1
r2
Fρ,1 Fρ,2
Fρ,net Fz,net
Figure 2: Optical trapping in an unfocused laser beam
Now, as laser beams refract off of the bead, they experience changes in their radial
momenta, pρ (momentum changes in the axial direction are the focus of the next sec-
tion and are not considered here). In particular, pρ becomes a non-zero quantity after
refraction. Thus, in order to restore the original momentum of the system, the bead
will itself experience an equal (yet opposite in direction) change in radial momentum.
If the laser has a Gaussian beam profile then the highest beam intensity will occur at
the centre of the beam profile, resulting in a net radial force Fρ,net which pulls the bead
into the centre of the laser beam (see Figure 2). This net force exists radially outwards
from the centre of the laser beam, creating the aforementioned gradient force potential.
Of course, the gradient force alone is not enough to trap a bead in three dimensions;
a force potential must exist in the axial direction as well. For this, we consider the
case of a focused laser beam.
2.1.2 Scattering Force
Although the gradient force potential is still present in a focused laser beam, an addi-
tional potential also exists in the axial direction, due to changes in the axial momentum
of laser photons.
Consider a microbead located at the radial centre, but not at the focal point, of
a focused laser. If the bead is located below the focal point of the laser, then pho-
tons will experience changes in both their axial and radial momenta as laser beams
refract off of the bead. In particular, pρ increases while pz decreases for all photons.
Since the focused beam is radially symmetric, the net radial momentum of the system
is unchanged after refraction. However, in order to conserve the axial momentum of
the system, the bead will have to compensate for the axial momentum loss of the pho-
tons; it experiences a force Fz,net which pushes it towards the focal point (see Figure 3).
3

focal point
Fz,net
r1
r2
Figure 3: Optical trapping below the focal point
of a focused laser beam
focal point
Fz,net
r1
r2
Figure 4: Optical trapping above the focal point
of a focused laser beam
Similarly, a bead which is located ahead of the laser focus will experience a force
Fρ,net which pulls it back towards the focal point (see Figure 4). This is due to the
fact that upon refraction off of the bead, laser photons will now experience an increase
in axial momentum. As before, changes in radial momentum are compensated for by
the radial symmetry of the beam.
Thus, a force potential also exists in the axial direction for focused laser beams (the
so-called scattering force potential). In practice, this force potential holds the trapped
bead slightly ahead of the focal point and not directly at the focal point itself.
2.2 Force Calibration
Due to the scattering and gradient forces, the focal point of a laser beam essentially
forms a Hookian spring potential for a trapped particle. If the bead is subject to an
external force, Fext, it will be displaced by ∆x according the relation (Fext = −ktr∆x).
Therefore, in order to use the OT to measure force, it is necessary to measure the “trap
stiffness” (ktr) of a given OT apparatus. The trap stiffness is equivalent to a spring
constant and is generally on the order of pN/nm.
Measurement of the trap stiffness, however, is not as straight forward as it would
be with a macroscopic mass on a spring. This is because the beads are very small
in size and are in a liquid solution at a certain temperature, T. They will therefore
be subject to a very large number of collisions per second with water molecules—thus
undergoing Brownian motion (essentially randomly changing direction over time). It
is therefore useful to consider the motion of the bead in the frequency domain. More
specifically, we consider the power spectral density, Sx(f), of the bead which can be
written as the squared magnitude of the Fourier transform of its time series[3]:
4

Sx(f) = |˜x(f)|2 (1)
Here, ˜x(f) is the Fourier transform of the position of the bead in the time domain.
We have just considered one dimension.
Because the system is highly over-damped, the mass of the bead is assumed not to
influence its motion for the frequencies of interest (f < 100kHz)[3]. The bead’s posi-
tion in one dimension over time (x = x(t)) can be modeled by the Langevin equation:
γ ˙x + ktrx = F(t) (2)
Here, γ is the hydrodynamic drag coefficient described by Stokes’ Law[4] (see Appendix
A for a more in-depth discussion), ˙x = dx(t)
dt and F(t) is the random force on the bead
from collisions which cause its Brownian motion. Assuming that this “noise” force
is completely random over time (made up of all frequencies–white noise), its power
spectrum is constant and can be given by:
SF (f) = 4γkBT (3)
Here, kb is Boltzmann’s constant. So, taking the Fourier transform of Equation 2, we
can write the power spectral density of the bead, Sx, as:
Sx(f) =
kbT
γπ2(f2
c + f2)
(4)
This is called a “Lorentzian” curve[3] where a characteristic corner frequency, fc, has
been defined according to ktr = 2πγfc.
The trap stiffness can then be measured by fitting Equation 4 to the measured power
spectrum of a trapped bead and determining the corner frequency of the fit.
While the power spectral density function of a trapped bead is well-modeled by Equa-
tion 4 for most applications, an additional correction must be made to the hydrody-
namic drag coefficient, γ, in order to account for effects which are experienced in the
vicinity of a surface.
γ =
6πηr

1 − 9
16
r
d

+ 1
8
r
d
3
− 45
256
r
d
4
− 1
16
r
d
5
(5)
Generally, γ is proportional to both the viscosity of the medium and the radius
of the bead. This correction is needed to account for the abrupt change in apparent
5

viscosity when a surface is encountered. It is described by Equation 5, where r is the ra-
dius of the bead, d is the distance to the surface, and η is the viscosity of the medium[5].
Figure 5: Typical power spectrum and Lorentzian fit
Figure 5 shows a typical power spectrum and corresponding Lorentzian fit for a
trapped bead. In this example, the corner frequency of the fit is fc = 246.2 ± 2.14 Hz.
6

3 Experimental Methods
3.1 Apparatus
Figure 6: The optical tweezer apparatus
A schematic diagram of the OT apparatus is provided in Figure 6, which depicts
the compression of confined DNA in a nanochannel. Laser power is controlled by the
combination of the half-wave plate (P1) and the first polarization beam-splitting cube
(PBS1). The quarter-wave plate (P2) just before lenses L1 and L2 which direct the
beam into the objective ensures that circularly polarized light enters the trap (other-
wise there would be radial asymmetry in the beam profile). The second beam-splitting
cube (PBS2) directs backscattered light (indicated by dotted red lines) to a spatial
filter (SF) placed in front of the quadrant photodiode (QPD). The QPD is a 4-channel
photodiode capable of measuring the displacement of the bead in 3 dimensions (this
is further described in Section 3.1.1). Trapping occurs at the focal point of the beam
near the microscope objective.
A 1064 nm Class IV CO2 laser with a maximum power of 2 W is used in the
apparatus (see Figure 6). The microscope used is a Nikon Ti2000 inverted microscope
with TIRF or Epi illumination modes. The objective is a 100x oil immersion objective
with a NA of 1.49, which is good for shallow trapping but less effective for deep trapping
due to the spherical aberration effect[5]. Other objectives such as water immersion are
good for deep trapping.
3.1.1 Quadrant Photodiode (QPD)
The QPD uses voltage differences between channels to measure the displacement of the
bead. Displacement in the ˆz direction is determined from the total voltage fluctuations
of all 4 channels, since the relative intensity of backscattered light from the trapped
7

bead is indicative of the relative distance from the objective in the ˆz direction. In the
ˆx and ˆy directions, this is accomplished by taking the voltage difference between pairs
of channels of the QPD, normalized by ˆz; the voltage difference between the right and
left pairs of channels gives the x displacement, while the voltage difference between the
top and bottom pairs of channels gives the y displacement.
2 1
3 4
Figure 7: Schematic illustration of the QPD
Thus, referring to Figure 7, z displacement is determined from channels 1, 2, 3,
and 4 combined. Meanwhile, x displacement is determined from the voltage difference
between channels 2-3 and 1-4, and y displacement is determined from channels 1-2 and
3-4. In order to accurately relate voltage differences to displacement, the backscattered
beam must be properly centered on the QPD. This alignment is performed manually
prior to any acquisition of data.
3.2 Procedure
The first step in calibrating the optical tweezers involves mixing a solution of fluores-
cent polystyrene microbeads in distilled water. For this purpose, a 1.5 µL sample of
a concentrated polystyrene microbead solution with bead diameter 0.4µm is diluted
with many parts of distilled water. In qualitative terms, a dilution of 1:100 parts pro-
duces a relatively concentrated solution, often resulting in multiple microbeads being
simultaneously trapped in the OT. Conversely, a dilution of 1:1000 parts results in a
fairly dilute solution which readily permits the trapping of a single bead at a time.
Once the solution has been prepared to a desired concentration, a 0.4 µL sample is
pipetted onto a microscope slide and mounted with a thin glass coverslip. The edges
of the coverslip are then sealed to the microscope slide with a thin layer of clear nail
polish in order to prevent air bubbles or leaks from producing unwanted solution flows
8

on the slide (this step also helps to prevent samples from becoming contaminated).
Furthermore, as the slides must be placed upside-down on the stage of the microscope,
the use of clear nail polish as a sealing agent also prevents unwanted movement of the
sample itself.
Once the sample is in place, the laser is turned on and a shutter opened which al-
lows the optical trap to form. An oil-immersion objective is used in order to view the
contents of the slide at high magnification/resolution (see Figure 8).
Figure 8: Fluorescent beads in solution as viewed from the microscope
Figure 8 shows the appearance of fluorescent beads when viewed through the mi-
croscope. It should be noted that the left-most of the four beads which are shown in
this image appears slightly blurred because it is located out of the focal plane of the
microscope. That is, it sits at a different height in the microscope slide than the other
three beads. Furthermore, the laser shutter is closed in this image, so no trapping is
taking place.
The movement of the microscope stage can be controlled either manually or by the
use of software routines. Without the use of any additional equipment, resolutions on
the order of 10−6
m are readily achievable. However, by mounting an external nanos-
tage on top of the existing platform, enhanced resolutions on the order of 10−9
m can
be achieved. Furthermore, it is possible to automate the movement of the nanostage
so that measurements can be taken at regular intervals. This permits more accurate
characterizations of optical trap stiffness to be made.
Once a bead is trapped, the QPD records the position of the bead in the ˆx, ˆy, and
ˆz directions over an interval of ∼ 30s.
9

Figure 9: A screen shot of the software which displays the QPD readout. Parameters such as
sample rate (red arrow) and voltage limits (purple arrows) can be set in this program.
The time series data of the trapped bead is recorded in the TimeSeriesStreaming
program (a screen shot of this program can be seen in Figure 9). The program outputs
a file which consists of four columns: one time column and three voltage columns (one
for x, y, and z). This file is then put into the MatLab Rroutine TweezerCalib which
uses the methods described in Section 2.2 to convert the voltage measurements (which
indicate position) into a power spectrum. Additionally, this program applies adjustable
noise-reducing filters to the data, increasing the signal-to-noise ratio. An output of this
program can be seen in Figure 5.
3.3 Analysis
Once data is acquired, it was analyzed using a purpose-built Matlab R routine [6].
This program converts the x and y positions of the bead into two power spectra (one
for each direction). A Lorentzian curve (Equation 4) is then fit to each data set and
the corresponding corner frequency is returned.
As might be expected, the quality of the Lorentzian fits produced using the Matlab R
program varies. In some cases, as for Figure 5 in Section 2.2, the program is able to
fit the data quite well. Fits of this quality are referred to as “excellent” fits. However,
in other cases, the results are not nearly as good and there are obvious differences
between the data and calculated fit (see Figure 10 for a “poor” fit). When situations
such as this arise, the measurement is omitted from the rest of the data set.
No numerical method was used to determine whether or not a fit was “good”.
Rather, the decisions were made qualitatively by comparing the fit to others of the
same data set. Of course, more rigorours methods such as the chi-squared test could
have been used, but such a robust method was not needed in our applications.
10

Figure 10: Data and corresponding calculated Lorentzian fit. There are large differences between
the data and fit over a wide range of frequencies.
4 Results and Discussion
4.1 Beam Profile
Early data sets which were collected showed drastic inconsistencies between the corner
frequencies as measured in the x and y directions (see Table 1 in Appendix B). This
was an issue because the circularly polarized beam is radially symmetric and should
therefore show consistent x and y corner frequencies. In the most extreme cases, these
differences could reach as high as 290%. Collectively, the x and y corner frequencies
differed by 105.22%. It should be noted that when the fitting parameters are unable
to generate an appropriate fit, complex corner frequencies result and are thus omitted
from the subsequent analysis.
These inconsistencies were attributed to asymmetry in the beam profile of the
backscattered light from the trapped bead. The laser emits a gaussian beam, so the
detected backscattered light should be detected by the 4-channel diode to be gaussian
as well. The profile appeared to be centered off-axis and contained other deviations
from the expected shape, possibly due to problems with the optical components of the
set up. This resulted in faulty detection of the beam by the 4-channel photodiode (see
Figures 11 and 12).
11

−3 −2 −1 0 1 2 3 4 5
0
0.5
1
1.5
2
x 10
4
Relative X−Position (arbitrary units)
Count
Figure 11: Beam profile in x direction, with mis-
aligned lens
−4 −2 0 2 4
0
0.5
1
1.5
2
x 10
4
Relative Y−Position (arbitrary units)
Count
Figure 12: Beam profile in y direction, with mis-
aligned lens
Eventually, a misaligned lens was found to be the cause of beam profile asymmetry.
Once this had been corrected, the beam profile was found to be greatly improved, with
a collective difference in the corner frequencies of the x and y directions of 28.98%
(see Table 1). The 4-channel also diode showed much more normal beam profiles (see
Figures 13 and 14).
Average Percentage Difference
in x and y Corner Frequency
Before correction After Correction
105.22 28.98
Table 1: Average values of the percentage difference in x and y corner frequencies, before and after
the correction of a misaligned lens which affected the beam profile
−0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25
0
0.5
1
1.5
2
2.5
3
x 10
4
Relative X−Position (arbitrary units)
Count
Figure 13: Corrected Beam profile in x direction,
with fixed lens
−0.15 −0.1 −0.05 0 0.05 0.1 0.15
0
1
2
3
4
5
x 10
4
Relative Y−Position (arbitrary units)
Count
Figure 14: Corrected beam profile in y direction,
with fixed lens
12

Figures 11, 12, 13, and 14 illustrate the beam profiles in the x and y directions,
before and after the correction of the misaligned lens. These figures are significant
because they represent how the QPD actually “sees” the beam profile. A qualitative
determination of beam profile quality can thus be obtained from the deviation between
the obtained data and the overlying gaussian profile which represents an ideal beam.
4.2 Corner Frequency vs. Bead Proximity to an Edge
Data collection as described in Sections 3.2 and 3.3 was done by varying the relative
position (in the z direction) of a trapped bead in a microscope slide with a coverslip
(see Figure 15). In order to find the upper edge, a trapped bead was raised by incre-
ments of 0.05 µm until it hit the top of the slide and was ejected from the trap. Then,
at this height, another bead was trapped and power series data was collected. More
data sets were taken every 0.5 µm until the bead hit the bottom edge (the coverslip).
A plot of corner frequency vs. relative trap position to edge can be seen in Figure 16.
upper edge (slide)
lower edge (coverslip)
z=0
y
xz
Figure 15: Diagram of trapped bead in slide. Position in z direction was varied
and x and y corner frequencies were measured at each height.
Looking at Figure 16, there is no clear relation between edge proximity and trap
stiffness in either direction. This set of data however is not a completely accurate
representation of the physics of our trap. This is because the points on this plot are
a result of the ability for the previously described Matlab program to fit a Lorentzian
Curve to the power series data. If the program did not fit the data properly, the re-
sulting corner frequency would not be accurate. This was the case for several of these
data points. In order to improve the fitting, certain parameters in the Matlab fitting
program such as frequency range and signal-noise ratio filters need to be adjusted to
optimize the fitting. Also, the variation of corner frequencies over a short change in
distance is quite large compared to the error bars (which are given by the Matlab pro-
gram) so it is necessary to determine the reason for that inconsistency in precision.
13

Figure 16: Diagram of trapped bead in slide. Position in z direction was varied
and x and y corner frequencies were measured at each height.
4.3 Corner Frequency as a Function of Time
Since no clear trend in trap stiffness was observed when varying the z-position of the
trap, the decision was made to take data from a trapped bead without varying its
position. The long-term nature of the trap could hence be better understood by pe-
riodically measuring the corner frequency of the trap at different points in time. In
other words, if some temporal trend in trap stiffness was discovered, it might explain
why there is no clear relationship between trap stiffness and proximity to an edge.
Furthermore, the identification of such a temporal trend would be important for future
applications of the trap.
Data taken for a bead at a fixed position is shown in Figure 17. There is no ap-
parent trend in trap stiffness with time, and, as before, there does not appear to be
any general correlation between x and y trap stiffness. Although the two quantities do
seem to follow one another in the range of 14 to 24 minutes, the overall discrepancies
in the data suggest that this is likely a coincidence. Furthermore, the large deviations
in y trap stiffness, particularly at 34 and 44 minutes, indicate that there is something
seriously wrong with either the trap itself or with the detection of the bead’s y position
by the QPD. Thus, this data cannot be used to determine the true stiffness of the trap.
It is worth mentioning that the circled data points in Figure 17 correspond to
Lorentzian fits which were subjectively determined to be “excellent” fits, as described
in Section 3.3. The average corner frequencies of these circled points are 279.83 ± 66.01
Hz and 329.86 ± 32.08 Hz in the x and y directions, respectively. It is also interesting
14

0 10 20 30 40 50 60 70 80
0
200
400
600
800
1000
1200
1400
Time (minutes)
CornerFrequency(Hz)
X
Y
Figure 17: Data taken for a trapped bead at a fixed position. Circles indicate “excellent” fitting
between the Lorentzian curve and the data.
to note that although the y data varies by a large amount, the circled corner frequen-
cies are quite close to the x corner frequencies from the same trial. This is what one
would expect in a radially symmetric optical trap.
Using only the circled points from Figure 17, we can apply Stokes’ Law (Equation 7
in Appendix A) to come up with estimates for value of the trap stiffness, ktr. The values
are tabulated in Table 2. Note that the associated uncertainties are not necessarily
meaningful, since these values were subjectively chosen from a larger data set. Rather,
the results serve to provide a rough idea of the properties of the trap.
x Trap Stiffness (pN/nm) y Trap Stiffness (pN/nm)
0.00591 ± 0.00139 0.00693 ± 0.00068
Table 2: Estimate of the optical trap stiffness in the x and y directions.
When the entire data set is considered, however, the time-averaged x corner fre-
quency was found to be 308.88 ± 53.23 Hz. Ideally, one would like to measure trap
stiffness very precisely, and an uncertainty of 100 Hz in the corner frequency is far
from ideal. In addition to the large inconsistencies in y corner frequency, this also
suggests that there is a systematic source of error in the OT apparatus.
In order to confirm whether this systematic error was due to the trap or to improper
detection of backscattered light by the QPD, the QPD was rotated by 90◦
. This ef-
fectively reversed the orientation of the x and y coordinates in the detection system
(recall that the QPD measures x and y position by taking the voltage difference be-
15

tween corresponding pairs of channels). Thus, if the x corner frequency now showed
discrepancies instead of the the y corner frequency, the QPD could not be the source
of the problem. Indeed, once the experiment was repeated, discrepancies between x
and y corner frequencies and large uncertainties in the time-averaged corner frequency
still remained. This result confirmed suspicions that the apparatus was the root of the
problem.
4.4 Backscattered Light Intensity vs. Bead Proximity to Edge
The purpose of this research is to use optical trapping in a nanochannel (whose width
is comparable to the diameter of the bead). Since our final results rely mostly on the
data from the backscattered light, it is of interest to observe the effects that proximity
to the microscope slide/coverslip will have on its intensity. In order to observe these
effects, an experiment similar to the one described in Section 4.2 was carried out, how-
ever instead of analyzing the frequency components of the X and Y motion of the bead,
only the time-averaged total voltage (which can be related to intensity) was recorded.
0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
3.5
4
QPDReading
Relative Z−Position (µm)
Without bead
With bead
Figure 18: Backscattered light intensity with and without a trapped bead as a function of z-position
Figure 18 shows the averaged voltage readings at each position for data taken with
both an empty trap and a trapped bead. The purpose of taking data with an empty
16

trap was to determine what the effect of the aqueous solution and glass slide or cover-
slip on the backscattered laser light is in general. The results seem intuitive, as a trend
was observed whereby the most reflection occurs at the surface of both the coverslip
and microscope slide. This can be observed from the two peaks of the blue data at
approximately 3µm and 25µm in Figure 18. With a trapped bead, data could not be
taken below approximately 12µm. This is likely due to the fact that the edge of the
coverslip pushes the bead out of the trap at this point.
In the region from 20µm to 25µm, the backscattered light from a trapped bead
decreases in intensity despite the fact that this region should correspond to the edge
of the microscope slide. It is not clear how the bead remains trapped at this point,
assuming that this is in fact where the microscope slide begins. This is perhaps due
to the scattering force pinning the bead against the slide even though it is no longer
located at the focal point. In this case, we might expect to detect less backscattered
light because of reflection off of the bead’s spherical surface (as opposed to the flat
surface of the microscope slide when the bead is absent). Indeed, this is observed in
Figure 18 (detailed views of this figure are provided in Appendix C).
These effects could be better understood by recording the backscattered light of a
bead moving through the focal point of the trap. This could be accomplished by align-
ing the trap with a bead which is stuck to the surface of the coverslip or microscope
slide (this occurs naturally occurs whenever a sample is prepared). The stuck bead
could then be moved through the focal point of the trap by adjusting the height of the
microscope stage (see Figure 19).
upper edge (slide)
focal planeincoming light
backscattered light
a. b. c.
Figure 19: Moving a stuck bead through the focal point of an optical trap.
Figure 19a shows a stuck bead being held above the focal point of the trap. Figure
19b shows the bead being brought through the focal point—here, we expect the most
backscattered light to be detected. Finally, Figure 19c shows the bead being brought
below the focal point.
17

5 Conclusion
Despite some inconsistencies, the apparatus has been providing results fairly consistent
with the theory. Specifically, the measured power spectra of trapped beads have in-
deed followed Lorentzian distributions—it is only a matter of reducing certain sources
of error in the optics of the apparatus before the OT can be used to accurately measure
forces. Even so, we are able to come up with rough estimates for the value of the trap
stiffnesses (ktr = 0.00591 ± 0.00139 pN/nm in the x direction and ktr = 0.00693 ±
0.00068 pN/nm in the y direction). This should hopefully be completed over the next
month or two. Additionally, even without calibration fully complete, the point has
been reached where experiments involving trapped DNA can begin.
For example, successfully trapping a bead and a single strand of DNA in a nanochannel
is not a trivial task. Developing an efficient method for creating this arrangement is
not something which depends on the trap being fully calibrated and can be started im-
mediately. Also, being able to precisely sweep a bead back and forth in a nanochannel
is not something that can be done manually. This procedure will involve some sort of
automation and will most likely require feedback to ensure the bead does not collide
with the channel walls.
Ultimately, the research done this semester has indicated that the goals outlined in
section 1 are certainly attainable in the near future. A. Stange will be continuing his
contribution to this research over the next few months and hopefully some conclusive
results will be obtained.
6 Acknowledgements
The authors wish to thank Professor Walter Reisner for providing the opportunity to
contribute to this exciting research. This project could not have been done without
his knowledge on the subject and suggestions for experiments. The authors would also
like to thank Ahmed Khorshid for providing training on how to use the infared laser
and other features in the apparatus. Ahmed also was able to address any issues that
came up during experiments. Also, the authors wish to thank Charles Bouchard who
had also been carrying out experiments for calibrating the OT and was able to provide
much advice on measurement techniques.
7 Author Contributions
A. Stange and R. Muller-Moran performed the experiments. A. Stange contributed to
the force calibration theory section, while R. Muller-Moran contributed to the optical
tweezers theory section. Both authors contributed to the experimental methods section.
R. Muller-Moran contributed to the results and discussion sections concerning beam
18

profile and corner frequency as a function of time, while A. Stange contributed to the
results and discussion sections concerning corner frequency as a function of time and
backscattered light intensity as a function of position. A. Stange also contributed to
several figures throughout this report.
References
[1] A. Ashkin, J. M. Dziedzic, J. E. Bjorkolm, and S. Chu, Observation of a single-beam
gradient force optical path for dielectric particles, Opt. Lett. 11, 288290 (1986).
[2] Ghislain L. P. and Webb W. W., Scanning-force microscope based on an optical
trap, Opt Lett., 18:19, 1678-1680 (1993).
[3] F. Gittes and C.F. Schmidt, Thermal noise limitations on micromechanical exper-
iments, Eur. Biophys. J. 27, 7581 (1998).
[4] F. Reif, Fundamentals of statistical and thermal physics. McGraw-Hill, NewYork,
1965.
[5] K. C. Vermeulen, G. J. L. Wuite, G. J. M. Stienen, and C. F. Schmidt. Optical
trap stiffness in the presence and absence of spherical aberrations. Optical Society
of America (2006)
[6] Tolić-Nørrelykke, Iva Marija et al. MatLab program for precision calibration of
optical tweezers. Computer Physics Communications 159 (2004) 225-240
[7] Press, William H., Teukolsky, Saul A., Vetterling, William T. and Flannery, Brian
P. Numerical Recipes: The Art of Scientific Computing. 3rd ed. New York: Cam-
bridge UP, 2007.
[8] Hughes, Ifan G. and Hase, Thomas P. A. Measurements and their uncertainties: a
practical guide to modern error analysis. Oxford: Oxford University Press, 2010.
[9] J. H. G. Huisstede, K. O. van der Werf, M. L. Bennink, and V. Subramaniam, Force
detection in optical tweezers using backscattered light, Opt. Exp. 13:4, 11131123
(1998).
[10] J. Bernstein, P. M. Fishbane, and S. Gasiorowicz, Modern Physics. Prentice Hall,
New Jersey, 2000
19

A Stoke’s Law
Stokes’ law is derived from the Navier-Stokes equations which apply Newton’s Second
Law to the motion of fluids. The result is only valid in the limit of low Reynolds num-
ber, a ratio which describes the importance of inertial and viscous forces to a flowing
system. The Reynolds number is given by Equation 6:
Re =
ρvL
η
(6)
where ρ is the density of the fluid, v is the relative velocity of an object in the fluid,
and L is the length travelled by the fluid (ie. the length of the object). Since the beads
are only moving under the influence of Brownian motion and have diameters on the
order of 10−7
m, the conditions of our project do indeed correspond to a low Reynolds
number. Thus, the hydrodynamic drag coefficient as given by Stokes’ Law is:
γ = 6πηr (7)
For water, with a viscosity of approximately 8.90 × 10−4
Pa·s at 25◦
C, the value of
γ is approximately 0.0168r Pa·s.
20

B Percent Differences Between x and y Corner Fre-
quencies
Relative Position (µm) Percentage Difference Between x and x Corner Frequency
1 -29.31
2 89.75
3 20.15
4 85.81
5 220.64
6 292.69
7 210.36
8 167.53
9 117.03
10 54.05
11 28.98
12 25.38
13 19.04
14 33.20
15 154.41
16 -100.00 - 4.5017i∗
17 78.80
18 105.75
19 160.42
20 196.21
21 89.48
22 43.21
23 151.34
Average Percentage Difference 105.22
Table 3: Percent differences of corner frequencies as measured in the x and
y directions with misaligned lens. Note that entries marked with an asterisk
are omitted from the analysis as they are aphysical
21

Relative Position (µm) Percentage Difference Between x and y Corner Frequency
1 -50.88
2 55.75
3 32.29
4 22.08
5 25.50
6 20.77
7 31.04
8 37.89
9 37.90
10 42.86
11 28.24
12 52.00
13 31.50
14 31.73
15 18.63
16 -7.79
17 5.41
18 -13.49
19 40.49
20 30.59
21 29.38
22 47.93
23 55.00
24 62.26
25 57.51
Average Percentage Difference 28.98
Table 4: Percent differences of corner frequencies as measured in the x and
y directions after misaligned lens is fixed. The symmetry of the beam pro-
file was found to be greatly improved once an important component of the
optical system was realigned.
22

C Backscattered Light Intensity vs. Bead Proximity
to Edge
0 5 10 15 20 25 30
0
0.5
1
1.5
2
2.5
3
3.5
4
QPDReading
0 5 10 15 20 25 30
RelativeNoise
X
Y
Z
Figure 20: A plot of total QPD voltage as a function of relative trap position (without a bead) in
an aqueous solution in microscope slide with a coverslip. Standard deviations from the mean are
shown at each height.
23

12 14 16 18 20 22 24 26 28
0
0.2
0.4
0.6
0.8
1
QPDReading
12 14 16 18 20 22 24 26 28
RelativeNoise
X
Y
Z
Figure 21: A plot of total QPD voltage as a function of relative trap position (with a bead) in
an aqueous solution in microscope slide with a coverslip. Standard deviations from the mean are
shown at each height.
24

10 12 14 16 18 20 22 24 26 28 30
0
0.5
1
1.5
QPDReading
With bead
Without bead
Figure 22: Close-up of the intersection between the QPD voltage readings, with and without a
trapped bead, as a function of relative trap position.
25

Final Project Report

More Related Content

What's hot (20)

Similar to Final Project Report (20)

Final Project Report