Spring 2011:Optical Trapping Lab

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20.345: Bioinstrumentation Project Lab

Team FTS

Huayu & Cory

XY position detector calibration

check to see the direction of scan (X)
check to see the direction of scan (Y)
Xcal - volt Vs. position center slope = -3.457
Ycal - volt Vs. position center slope = m = -7.816

Roll Off

Power of laser 10mW, Calculated stiffness =9.28E-6 N/m = .00926 pN/nm




for 10 mW : 2.09*10^-9 N/m

Team Humble

XY position detector calibration


Trap stiffness by equipartition theorem

Method1 stiffness.jpg

These values are off by about 8 orders of magnitude. This graph is up temporarily until I figure out what is going wrong with my calculations.

Trap stiffness by PSD rolloff

Method2 stiffness.jpg

These values are about 2 orders of magnitude larger than those found in the OTKB document.

Alternative Attempt for Team Humble

(by Emily)





Not only is this last method trending the wrong way, but it is many orders of magnitude off from the previous two. This could be due to errors in the data collection or (and more likely) errors in data manipulation calculations.

Trap stiffness by Stokes' theorem

Method3 stiffness.jpg

These values are about the correct order of magnitude, but the slope of the graph should be positive, not negative! Again, this graph is up temporarily until I can figure out what data processing mistakes are going on.

Attempt #3 for Team Humble

by Kristin

The goal of this lab was to measure the effective spring constant ("trap stiffness") of the optical trap system.

3 samples were taken: 10mW, 20mW, and 30mW. All graphs shown are for the 10mW sample. Similar manipulation was done for the 20 and 30 mW samples, and those results are shown at the end.

XY Scan:


From this we learned that the the optical trap scanned past the center of the bead at t = 861 time units. It is the time when the bead moved the least along the X axis (red) for the most amount of drive on the Y axis (blue). That is, if we are oscillating in the exact center of the bead, then dragging the bead up-and-down should not move it left-and-right very much.

Determine calibration constant:


Here we learn the conversion between volts and distance for our particular system. The slope of this line = QPD volts / stage movement volts. The stage has been determined to move 2.22e-6 meter/volt. so slope/2.22 = QPD volt/um.



So one of the major forces that the optical trap must overcome to move the bead, is the drag force from the surrounding fluid. According to Stokes' law, F = 6πrv*visoscity = kx. A line was fit to this to estimate k: .3148e-3. The magnitude was kept positive to maintain a consistent sign convention.

Equipartition Theorem:

According to the equipartition theorem,
(3/2)kBT = thermal energy

When holding a bead in place, it is these thermal motions that the optical trap must counteract. Thus,
(1/2)kx2 = potential energy from OT "spring" = (3/2)kBT

We solve for k, and find that k = 2.315e-07.

PSD roll-off: The final method for estimating the spring constant uses the power spectral density of the bead's thermal motion. It turns out that the corner frequency (fo) of the power spectrum is related to the trap stiffness:

k = 2πfoβ
where β = 3πηd, d = bead diameter, η = viscosity of medium.

We find fo by fitting the form of the power spectrum to the actual power spectrum:


fo can be extracted from the form.

We do so, and find that k = 1.834e-05.


10mW 20mW 30mW
Stokes-Drag 3.148e-04 4.508e-04 3.550e-04
Equipartition 2.315e-07 5.1e-09 2.141e-07
PSD roll-off 1.834e-05 1.613e-05 2.907e-05


Alas, the results do not look great. Not only are the different methods orders of magnitude away from each other, but also the 10mW, 20mW, and 30mW do not increase proportionally as expected. One explanation is that the bead may not have been completely attached to the coverslip. This would have contributed additional forces and made the Stokes-Drag estimation too large, and indeed the S-D coefficient is much larger than the other two. The equipartition method, likewise, is vulnerable to other factors that may influence a trapped bead's motions, such as a slight flow in the medium. Failure to include these methods would make this method's trap stiffness too small - and indeed, the kequipartition is the smallest of the three. The PSD roll-off method is the most accurate, because it does not make assumptions about what is causing the bead's motions - it merely records them and extracts information from them. However, it is still an approximation, and still susceptible to poor fitting or other errors. Despite this, it is still the best estimate of the trap stiffness.

Team Watson (Drago)

XY position detector calibration

Sens x.jpg Sens y.jpg

Although the Y axis plot looks OK, I'm not sure of what happened to the X axis one

Trap stiffness by equipartition theorem

Eq x.jpg Eq y.jpg

The X-axis plot follows the expected trend of higher stiffness with higher power, but the Y axis has a very off-line measurement, although the orders of magnitude seem right.

Trap stiffness by PSD rolloff

PSD x.jpg PSD y.jpg

Both of these plots are right around the expected value, although they are around twice the values found for the Equipartition method (maybe there are some conversion errors).

Trap stiffness by Stokes' theorem

Stokes x.jpg Stokes y.jpg

Finally, this method seems to follow the expected trend, but its values are around an order of magnitude off from the previous methods. Again, this may be due to unit conversion issues (currently revising).

Team Watson (Emmanuel)

XY position detector calibration

QPD eq.jpg

  • QPD_XY[V/um] = 3.38e-4 * Power[mW] + 1.6e-3
  • QPD_YX[V/um] = 4.60e-4 * Power[mW] + 8.5e-3

Trap stiffness by equipartition theorem

Equipart eq.jpg

  • Equipart_stiff for X axis: kg/s^2 = 4.57e-16 * Power[mW] + 5.14e-16
  • Equipart_stiff for Y axis: kg/s^2 = 3.37e-16 * Power[mW] + 3.85e-16

Trap stiffness by PSD rolloff


  • PSD roll-off stiff for X axis: kg/s^2 = 1.8e-15 * Power[mW] - 51.6e-15
  • PSD roll-off stiff for Y axis: kg/s^2 = 1.9e-15 * Power[mW] - 60.6e-15

Trap stiffness by Stokes' theorem


Team with Brian, Siv, Gustavo

The following anaOpticalTrapCalibBross.jpglysis is done with data collected by "Team Watson" unless noted otherwise.

Power Spectral Density roll-off

Here, we use the free bead, fixed stage data from "Team GustavSiv".

Note that the corner frequency is approaching the Nyquist frequency and is thus occluded for higher laser intensities
Trap stiffness by PSD roll-off method. The super-linear trend is almost definitely attributed to the error in estimating the corner frequency for high laser intensities.

QPD Responsivity

To perform the next methods, it is necessary to calibrate the QPD.

Stuck bead fit x 10mW.png

Time-series waveform. The star indicates the selected subset to fit.


The procedure depicted in the above graphs is repeated for each measurement, in each of the x and y axes.

Don't be alarmed by the negative slope. The magnitude is positive!

===QPD Responsivity=== (another analysis) This analysis is using the same data to generate the QPD responsivity again. Determining Direction of Scan (XY) - X direction


Determining Direction of Scan (XY) - Y direction


Determining Direction of Scan (YX) - X direction


Determining Direction of Scan (YX) - Y direction


QPD Responsivity vs Laser Power


Equipartition Method

Equipartition methodGG.png The equipartion method gives results which are off by several orders of magnitude from the expected trap stiffness.

Navier-Stokes Drag Method

Better news here. The stage is swept back and forth and the amount of displacement of a free (as in not adhered to the cover slip, but trapped by the laser) bead is plotted over stage velocity.

Navstok x 30mW.png OpticalTrap1Blross.jpg

The best fit line to the data gives a measure of trap stiffness. If the trap were very stiff, then the particle would move very little while the fluid moves around it.

The procedure depicted above is repeated for several laser intensities and for each stage direction.

Navier stoke drag methodGG.png

Plot legend: green is stiffness in x and blue is stiffness in y.

OpticalTrap2 Navier bross.jpg

The point at 70mW unfortunately breaks the expected linear trend.

Team Lai

Position Calibration

Xpos.png Ypos.png

Force Calibrations

Three measurements taken at 30, 60, 90 mW. Error bars show standard error.

Equipartition Method

Xeq.png Yeq.png

PSD Rolloff Method

Stokes Method

Xstokes.png Ystokes.png

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