Download Lecture 14 (3/06/13) "Optical Traps III"

Mid-term next Monday: in class
(bring calculator and a 16 pg exam booklet)
TA: Sunday 3-5pm, 322 LLP:
Answer any questions.
Today: Fourier Transform
Bass (or Treble Booster)
Make Optical Traps more sensitive
Improve medical imaging (Radiography)
10 minute tour of Optical Trap
Optical Traps (Tweezers) con’t
Dielectric objects are attracted to the center of the beam, slightly
above the beam waist. This depends on the difference of index of
refraction between the bead and the solvent (water).
Vary ktrap with laser intensity such that ktrap ≈ kbio (k ≈ 0.1pN/nm)
Can measure pN forces and (sub-) nm steps!
http://en.wikipedia.org/wiki/Optical_tweezers
Requirements for a quantitative optical trap:
1) Manipulation – intense light (laser), large gradient
(high NA objective), moveable stage (piezo stage) or
trap (piezo mirror, AOD, …) [AcoustOptic Device- moveable laser pointer]
2) Measurement – collection and detection optics
(BFP interferometry)
3) Calibration – convert raw data into forces (pN),
displacements (nm)
Brownian motion as test force
≈0
..
mx + g x + kx = F(t)
Inertia
term
(ma)
Langevin equation:
kBT
Trap force
Drag force
Fluctuating
γ = 6πηr
Brownian
Inertia term for
um-sized objects
is always small
(…for bacteria)
force
<F(t)> = 0
<F(t)F(t’)> = 2kBTγδ (t-t’)
kBT= 4.14pN-nm
Autocorrelation function x(t )x(t )
ΔtΔt Δt
x(t )x(t )
Autocorrelation function x(t )x(t )
ΔtΔt Δt
x(t )x(t )
Why does tail become wider?
Answer: If it’s headed in one direction, it tends to
keep going in for a finite period of time.
It doesn’t forget about where it is instantaneously.
It has memory.
x(t )x(t )
<F(t)> = 0
<F(t)F(t’)> = 2kBTγδ (t-t’)
This says it has no memory.
Not quite correct.
Brownian motion as test force
Langevin equation:
g x + kx = F(t)
Exponential autocorrelation function
k BT  k t  t 
x(t )x(t ) 
e
k
FT → Lorentzian power spectrum
4k BT
1
Sx  f  
2
2
k 1   f fc 



= Ns/m
K= N/m
Notice that this follows
the Equilibrium Theorem
x
2
Corner
frequency
fc = k/2π
k BT

k
kT=energy=Nm
S= Nm*Ns/m/ (N/m)2
= m2sec = m2/Hz
4kBTg
1
Sx ( f ) =
2
2
k 1+ ( f fc )
As f 0, then
4kBTg
Sx ( f ) =
k2
As f fc, then
4kBTg 1
Sx ( f ) =
k2 2
As f >> fc, then
Sx ( f ) ® 0
kB T
Sx ( f ) = 2 2
p f
Langevin Equation FT: get a curve that looks like this.
1. Voltages vs.
time from
detectors.
2. Take FT.
3. Square it to
get Power
spectrum.
4. Power
spectrum = α2 *
Sx(f).
Power (V2/Hz)
Determine, k, a   6hr
fc 
4kBTg
k 2a 2
Note: This is Power spectrum for
voltage (not Nm)
kT=energy=Nm
S= Nm*Ns/m/ (N/m)2
Sx(f)= m2sec = m2/Hz
Power spectrum of voltage
Nm V divide by a2.
k BT
p 2g f 2a 2
Frequency (Hz)
k
2
What is noise in measurement?.
The noise in position using equipartition theorem
 you calculate for noise at all frequencies (infinite bandwidth).
For a typical value of stiffness (k) = 0.1 pN/nm.
<x2>1/2 = (kBT/k)1/2 = (4.14/0.1)1/2 = (41.4)1/2 ~ 6.4 nm
6.4 nm is a pretty large number.
[ Kinesin moves every 8.3 nm; 1 base-pair = 3.4 Å ]
How to decrease noise?
Power (V2/Hz)
Reducing bandwidth reduces noise.
fc 
4 k BT 2
a
k2
k BT
 2f 2
k
2
a2
Frequency (Hz)
If instead you collect data out to a
lower bandwidth BW (100 Hz), you get
a much smaller noise.
Noise = integrate power spectrum over
frequency.
If BW < fc then it’s simple integration
because power spectrum is constant,
with amplitude = 4kBT/k2
Let’s say BW = 100 Hz: typical value of  (10-6 for ~1 mm bead in water).
But (<x2>BW)1/2 = [∫const*(BW)dk]1/2= [(4kBT100)/k]1/2 =
[4*4.14*10-6*100/0.1]1/2
~ 0.4 nm = 4 Angstrom!!
Basepair Resolution—Yann Chemla @ UIUC
3.40
1bp = 3.4Å
1
unpublished
2
3
1
2
2.04
4
3
5
4
1.36
5
6
6
0.68
7
7
8
UIUC - 02/11/08
0.00
0
2
Probability (a.u.)
Displacement (nm)
2.72
4
6
Time (s)
8 9
9
3.4 kb DNA
8
10
0.00
0.68
1.36
2.04
Distance (nm)
2.72
F ~ 20 pN
f = 100Hz, 10Hz
Observing individual steps
Motors move in discrete steps
Kinesin
Step size: 8nm
Asbury, et al. Science (2003)
Detailed statistics on
kinetics of stepping
& coordination
Can add more “base” or
treble to music.
Fig. 1.25: Illustration of the addition of sine
waves to approximate a square wave.
http://en.wikibooks.org/wiki/Ba
sic_Physics_of_Digital_Radio
graphy/The_Basics
1st two Fourier components
http://cnx.org/content/m32423/latest/
Fig.2
http://www.techmind.org/dsp/index.html
1st 3 components (terms)
1st 11 components
The representation to include up to the eleventh harmonic. In this
case, the power contained in the eleven terms is 0.966W, and
hence the error in this case is reduced to 3.4 %.
Filtering as a function of wavelength
Test your brain: What does the Magnitude
as a function of Frequency look like for the
2nd graph?
Can add more “base” or
treble to music.
Fig. 1.25: Illustration of the addition of sine
waves to approximate a square wave.
http://en.wikibooks.org/wiki/Ba
sic_Physics_of_Digital_Radio
graphy/The_Basics
A simple Radiogram: Enhanced Resolution by FFT
1.23: A profile plot for the yellow line
indicated in the radiograph.
Can think of spectra as the intensity as a function of
position or a function of frequency.  Fourier Transforms
http://en.wikibooks.org/wiki/Basic_Physics_of_Digital_R
adiography/The_Basics
A fundamental feature of Fourier methods is that they can be used to
demonstrate that any waveform can be approximated by the sum of a
large number of sine waves of different frequencies and amplitudes. The
converse is also true, i.e. that a composite waveform can be broken into
an infinite number of constituent sine waves.
2D spatial Filter with Fourier Transforms
Fig. 1.27: 2D-FFT for a wrist radiograph showing increasing
spatial frequency for the x- and y-dimensions, fx and fy,
increasing towards the origin.
http://en.wikibooks.org/wiki/Basic_Physics
of_Digital_Radiography/The_Basics
(a) Radiograph of the wrist.
(b) The wrist radiograph
processed by attenuating
periodic structures of size
between 1 and 10 pixels.
(c): The wrist radiograph
processed by attenuating
periodic structures of size
between 5 and 20 pixels.
(d): The wrist radiograph
processed by attenuating
periodic structures of size
between 20 and 50 pixels.
Class evaluation
1. What was the most interesting thing you learned in class today?
2. What are you confused about?
3. Related to today’s subject, what would you like to know more about?
4. Any helpful comments.
Answer, and turn in at the end of class.