Section #: Methodology
Due to the complexity of the apparatus and experimental procedures, this section will be broken up into the
following subsections:
#.1 – The Apparatus:
#.1.1 – Overview
#.1.2 – The Optical Track
#.1.3 – The Objective Stage
#.1.4 – Calibrating the Objective Stage
#.1.5 – The Quadrant Photodiode
#.2 – Experimental Methods
#.2.1 – Power Spectral Density
#.2.2 – Equipartition
Section #.1.1: Overview of the Apparatus
The primary apparatus used was the Thorlabs OTKB Optical Trapping System, and is shown in Figure #. The
apparatus is comprised of three primary modules including the laser source and control, optical track, and objective
stage. The laser source is a 975 nanometer class IIIB laser diode, which is fed into the optical track via a fiber optic cable.
The laser is maintained by a temperature controller and a 0~300mW power supply unit. The optical track (shown as the
vertically oriented central module in Figure #) and objective stage (far left of Figure #) are described in detail in the
following sections.
Figure #: Thorlabs OTKB Optical Trapping
System [A]. The apparatus is comprised of
three main modules. From left to right they
are: the objective stage, optical track, and
laser control module. The strain gauge and
piezoelectric controllers are also shown,
mounted vertically to the right of the optical
stage.
Section #.1.2: The Optical Track
The 975nm Laser provided by the laser control module enters the optical track via a fiber optic cable as shown in
Figure #. The function of the optical track is to focus the laser into the sample mounted to the objective stage, while also
allowing the sample to be viewed and recorded directly. To accomplish this, dichroic mirrors are used to direct the laser
through the apparatus. The distinct advantage of dichroic mirrors is that they strongly reflect specific wavelengths of
light, while allowing other wavelengths to pass through unimpeded [B]. When the experimental region is illuminated by
an LED, the dichroic mirrors allow light in the visible spectrum to reach the CCD camera while preventing the potentially
damaging laser light from entering the camera as well. An infrared filter further ensures that the camera will not be
damaged while viewing the optical trap. The end result is a view of the optical trap which is coaxial with the laser
creating the trap.
Illumination source
Dichroic mirror
QPD
Condenser lens
Objective lens
Stage
Dichroic
mirror
IR filter
CCD
camer
a
mirror
Figure #: Schematic view of the optical
track. The laser enters the track via
fiber optic cable and is deflected by a
series of mirrors. The objective lens
focuses the laser into a point within
the sample to create the optical trap.
The laser light is recollected by the
condenser and directed towards the
QPD.
Laser
mirror
The series of mirrors encountered by the laser have an added function of broadening the beam, allowing it
subsequently be focused into a sharp point by the objective lens. The divergent light is then recollected by a condensing
lens, and directed towards a quadrant photodiode (QPD), which is described in more detail in the following sections. As
shown in Figure #, an LED light source illuminates the region of the optical trap, and is subsequently observed by a CCD
camera. The camera used in this experiment was a Thorlabs UC480 camera which, along with its accompanying
software, interfaced with a computer terminal to facilitate the viewing and recording of the experiment.
Section #.1.3: The Objective Stage
A Thorlabs MAX-311D Translating Stage acts as the objective stage for this experiment. A slide of microspheres,
prepared from a diluted stock solution, is mounted to the objective stage for viewing. Immersion oil between the
objective lens and sample slide prevents the sudden change in refractive index the laser would otherwise experience
upon entering open air. This prevents unwanted refraction of the laser when exiting the objective lens, reducing
intensity and making the formation of an optical trap possible. The stage itself is translated by three piezoelectric
motors, which are able to translate the stage by approximately 20 µm along each of the three primary axes. The
piezoelectric motors are regulated by external strain gauges which provide closed-loop feedback to the motors. The
strain gauges minimize hysteresis and dielectric creep [D], and are directly controlled by interfacing with the Labview
Software on a computer terminal. Output from strain gauges is a direct measurement of the displacement of the
piezoelectric motors, within an unknown constant factor. In order to make precise measurements of the optical trap,
this constant would need to be determined by calibrating the motion of the objective stage.
Section #.1.4: Calibration of the Objective Stage
Calibrating the motion of the objective stage occurred in two steps: First, the UC480 camera was used to
correlate motion in the video stream to motion in real space, which then allowed strain-to-nanometer conversion factor
to be determined.
To correlate motion viewed on the UC480 video stream to actual displacement in real space, a micrometer slide
was mounted to the objective stage. The UC480 software was used to save a snapshot of the micrometer slide, which
was subsequently uploaded into a separate program called ImageJ. The ImageJ software allows for precise
measurement in still images or image sequences, and in our case allowed for a precise count of pixels along a straight
line between each of the marks on the micrometer slide. This allowed us to determine that the UC480 camera had a
resolution of approximately 45 nanometers per pixel.
Once the pixel-to-nanometer conversion factor was determined, it became possible to connect strain gauge
output to actual motion. A sample slide of microspheres was prepared and mounted to the objective stage for
observation, and a stationary microsphere was located within the sample. The Labview software was used zero out the
strain gauge readings. This effectively established a strain of 50% to be the middle of the piezoelectric motor’s range of
motion for each of the x- and y-axis directions. Once the strain gauges were zeroed out, Labview was configured to
oscillate the objective stage five times along the x-axis by 5% strain, while the UC480 software was set up to record an
image sequence of these oscillations. Upon completing the oscillations, the recorded image sequence was imported to
the ImageJ software for analysis. A pixel count between the left and right most maxima for each oscillation allowed a
distance-to-percent-strain conversion factor to be established. This process was repeated for multiple percent strain
amplitudes, as shown in Figure #, and a final conversion factor of 262 nanometers per Percent Strain was found.
Figure #: Scatter plot of the displacement of
a microsphere as a function of percent
strain. A linear fit of the data yields a
conversion factor of 262 nm/strain.
Section #.1.5: The Quadrant Photodiode (QPD)
Due to diffraction around a microsphere, the Brownian motion of a bead confined to an optical trap will partially
deflect the laser which creates the trap. The use of a quadrant photodiode allows these deflections to be observed and
collected as data, which is particularly useful for characterizing the Brownian motion of the microsphere. Figure # shows
a schematic view of how the QPD extracts position data from incident laser light [D].
Figure #: Schematic view of a QPD. Each
quadrant reports a different voltage due to
the incident laser light, and changes in the
laser’ position on the QPD can be related to
real-space displacement.
The QPD consists of a photosensitive material which is segmented into 4 quadrants. As laser light is incident on each of
these quadrants, they each report a particular voltage reading. As the incident laser is deflected by a trapped
microsphere, it changes the voltage reading from each of these quadrants, and subsequently be extrapolated into a
measurement of the real-space displacement of the trapped bead. However, since the voltages reported by the QPD are
arbitrary, the QPD will also need to be calibrated by finding a QPD Voltage to nanometer conversion factor. This
calibration step is done through different means in each of our experimental steps.
Section #.2: Experimental Methods
Once initial calibration of the experimental apparatus was completed, we were able to approach the task of
measuring the spring constant of the optical trap created by our laser. We measured the trap stiffness in two ways: A
power spectral density analysis, and an application of the equipartition theorem. The latter case acts primarily as a check
for the results from the PSD analysis.
Section #.2.1: Power Spectral Density Analysis
In order to collect motion data on a trapped microsphere, a sample slide of microspheres was prepared from a
10,000x diluted stock solution. The slide was mounted to the objective stage, along with immersion oil. The UC480
camera and viewing software were used, along with the manual adjustment knobs on the objective stage, to focus the
video feed into the sample. Manual traversal of the sample via the objective stage’s adjustment knobs allowed
microspheres to be located. Once an isolated, free moving microsphere was located, the laser control unit was turned
on and set near the maximum intensity of 300mW. Minor adjustment of the objective stage allowed the desired
microsphere to be pulled into the optical trap created by the laser.
In order to generate the Power Spectral Density the QPD was used to track the bead’s position within the trap as
it moved due to Brownian forces. Before collecting data Labview was used to monitor the QPD’s voltage readings, which
rendered the laser’s location as a point in a two-dimensional XY-plane. The QPD’s alignment was adjusted to bring the
laser’s reported location near the origin. For this alignment step, the maximum laser output was required to minimize
the fluctuations in the QPD voltage while trying to center the laser at the origin. After alignment was completed,
Labview was configured to sample the QPD voltage at a rate of 10 kHz for five seconds. The resulting set of data points
was saved to a file for later analysis, and the procedure was repeated for multiple laser intensities. An analysis of the
data was conducted via Mathematica, and reveals both the spring constant of the optical trap and the QPD calibration
factor.
Section #.2.2: Determining the spring constant via Equipartition
In order to try and verify our results from Power Spectral Density analysis, we collected an independent data set
to analyze with the Equipartition Theorem. The basis of this method will again involve the Brownian motion of a trapped
bead, so a calibration of the QPD will be necessary. To calibrate the QPD, a sample slide of microspheres was prepared
from the diluted stock solution, and mounted to the objective stage. After focusing the UC480 camera and locating a
free moving bead, the laser control unit was turned on and the bead was allowed to fall into the optical trap. The QPD
was zeroed out in Labview as described in the preceding section.
Once the QPD was zeroed out, the laser control unit was turned off, and the formerly trapped bead was allowed
to move freely. Once the free bead was clear of the optical trap the objective stage was moved until a stationary bead,
which was stuck to the glass slide, was located. After turning the laser back on to approximately 300mW and allowing
Labview to zero out the strain gauges controlling the objective stage, the stuck bead was carefully maneuvered into the
center of the trap. Since the bead was no free to fall into the trap automatically the objective stage was adjusted until
the QPD voltage once again settled near the origin, signifying that the stuck bead was properly centered. A 6% strain
oscillation was initialized in Labview for each of the x and y axes, and the resulting data was saved into a file for analysis.
Since the strain to displacement conversion factor was already known, this method provides a direct measurement of
the QPD voltage versus displacement.
After determining the QPD calibration factors for the x and y directions, they were applied to the QPD data
collected at the same laser intensity in Section #.2.1. This data set now reflected the absolute position of the trapped
bead as a function of time. In order to apply the Equipartition Theorem, this data set was converted from absolute
position to displacements in each of the x and y directions. The variance of this new displacement data set is directly
connected to the spring constant of the optical trap by the Equipartition Theorem. The results from this method are
compared to those gleaned from Power Spectral Density analysis in the following section.
Section 5: Data and Analysis
Data for the Power Spectral Density analysis was uploaded into Mathematica and separated into x and y axis
components. An application of the PeriodogramArray function, with imposed HammingWindow binning, produces a
discretized Fourier transform of the x and y components. A fit curve is applied to the data by passing the frequency
domain values into equation (#) with fit variables 𝜌 and 𝑓0. By iteratively adjusting the initial guess for each of these
variables, the curve can be fit to the graph of the Fourier transform. This allows the QPD conversion factor, 𝜌, and corner
frequency, 𝑓0, to be extracted. The stiffness of the optical trap can then be determined from the corner frequency by
using equation (#). The results of this analysis for four different laser intensities are summarized in Table # below. An
example fit graph produced by Mathematica is shown in Figure # as an example. The remaining graphs are all
qualitatively identical.
Table 1: Summary of Results from Power Spectral Density Analysis
nm
nm
pN
𝝆𝒙 (
)
𝝆𝒚 (
)
Laser Intensity
𝒌𝒙 ( )
QPDvolt
QPDvolt
nm
296 mW
700
713
0.020
pN
𝒌𝒚 ( )
nm
0.020
200 mW
881
871
0.016
0.016
125 mW
864
862
0.015
0.016
100 mW
691
705
0.022
0.022
Figure # shows the results from applying the Power Spectral Density to the Fourier transform of the position data. The
red curve represents the PSD fit function which is superimposed on the transformed QPD data, shown in blue. The
frequency domain plateaus prior to hitting the corner frequency, at which point the red curve bends downwards.
0.010
0.001
Figure #: Superposition of the PSD fit
curve (red) on the Fourier transform
of the QPD position data. The corner
frequency corresponds to the sudden
end of the frequency plateau.
10 4
10 5
10 6
10 7
10 8
1
10
100
1000
10 4
In the process of confirming our results from the Power Spectral Density, we determined a conversion factor for
QPD voltage to displacement for a laser intensity of 296 mW. Figure # shows graphs of the QPD voltage versus strain
nm
found in Section 4.2.1. A linear fit of the data about the origin reveals a conversion factor of 2770 (QPD
volts
nm
and 2994 (QPD
volts
) in the x-axis,
) in the y-axis, which are each approximately four times larger than the conversion factors found via
Power Spectral Density analysis. Since these values agree fairly well, we conclude that at least our QPD conversion
factors are consistent.
Figure #: Graphs of QPD voltage versus percent strain for the x- and y-axis QPD data, acquired for a laser
intensity of 296mW. Note that the QPD was set up to register x-axis motion as a change in the yDiff voltage.
The positions from the QPD data were converted into displacements along each axis by applying the conversion factors
found from Figure #. The variance of this new data set is related to the trap stiffness by equation (#), and yielded a
pN
pN
spring constant of 0.59 (nm) in the x-direction and 0.73 (nm) in the y-direction. These values are a factor of 30 larger
than those found through Power Spectral Density, and unfortunately make a poor justification for our results in the
previous section.