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

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). Can measure pN forces and (sub-) nm steps! Vary k trap with laser intensity such that k trap ≈ k bio (k ≈ 0.1pN/nm) Optical Traps (Tweezers) con’t

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 Drag force γ = 6πηr Fluctuating Brownian force Trap force = 0 = 2k B Tγδ (t-t’) kBTkBT k B T= 4.14pN-nm Langevin equation: Inertia term (ma) ≈0 Inertia term for um-sized objects is always small (…for bacteria)

ΔtΔt ΔtΔtΔtΔt Autocorrelation function

ΔtΔt ΔtΔtΔtΔt

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. = 0 = 2k B Tγδ (t-t’) This says it has no memory. Not quite correct.

Brownian motion as test force FT → Lorentzian power spectrum Corner frequency f c = k/2π  Exponential autocorrelation function Notice that this follows the Equilibrium Theorem Langevin equation:  = Ns/m  K= N/m kT=energy=Nm S= Nm*Ns/m/ (N/m) 2 = m 2 sec = m 2/ Hz

As f  0, then As f  f c, then As f >> f c, then

Power (V 2 /Hz) Frequency (Hz) 1. Voltages vs. time from detectors. 2. Take FT. 3. Square it to get Power spectrum. 4. Power spectrum = α 2 * S x (f). Determine, k,  r  Langevin Equation  FT: get a curve that looks like this. Note: This is Power spectrum for voltage (not Nm) kT=energy=Nm S= Nm*Ns/m/ (N/m) 2 S x (f)= m 2 sec = m 2/ Hz Power spectrum of voltage Nm  V divide by  2.

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. 1/2 = (k B T/k) 1/2 = (4.14/0.1) 1/2 = (41.4) 1/2 ~ 6.4 nm What is noise in measurement?. 6.4 nm is a pretty large number. [ Kinesin moves every 8.3 nm; 1 base-pair = 3.4 Å ] How to decrease noise?

Reducing bandwidth reduces noise. But ( BW ) 1/2 = [∫const*(BW)dk] 1/2 = [(4k B T  100)/k] 1/2 = [4*4.14*10 -6 *100/0.1] 1/2 If instead you collect data out to a lower bandwidth BW (100 Hz), you get a much smaller noise. Power (V 2 /Hz) Frequency (Hz) Noise = integrate power spectrum over frequency. If BW < f c then it’s simple integration because power spectrum is constant, with amplitude = 4k B T  /k 2 Let’s say BW = 100 Hz: typical value of  (10 -6 for ~1  m bead in water). ~ 0.4 nm = 4 Angstrom!!

3.4 kb DNA F ~ 20 pN f = 100Hz, 10Hz 1bp = 3.4Å UIUC - 02/11/08 Basepair Resolution—Yann UIUC unpublished

Kinesin Asbury, et al. Science (2003) Step size: 8nm Observing individual steps Motors move in discrete stepsDetailed statistics on kinetics of stepping & coordination

Fig. 1.25: Illustration of the addition of sine waves to approximate a square wave. sic_Physics_of_Digital_Radio graphy/The_Basics Can add more “base” or treble to music.

1 st two Fourier components

Fig.2

1 st 3 components (terms)

1 st 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 2 nd graph?

Fig. 1.25: Illustration of the addition of sine waves to approximate a square wave. sic_Physics_of_Digital_Radio graphy/The_Basics Can add more “base” or treble to music.

1.23: A profile plot for the yellow line indicated in the radiograph. adiography/The_Basics Can think of spectra as the intensity as a function of position or a function of frequency.  Fourier Transforms A simple Radiogram: Enhanced Resolution by FFT

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, f x and f y, increasing towards the origin. 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.