Optical Tweezers and their calibration

How Trap works 1

How Trap works 2

Calibration Methods 1. Histogram of Bead Positions 2. Power Spectrum of Bead Positions 3. Stokes Flow or Stokes Friction

Histogram of Bead Positions 1 a. Motivation and preliminary calculations. Idea : If the bead in trap makes a Brownian motion, The histogram of positions should be a gaussian !!! Application : The width of the gaussian is related with the stiffness of trap. Math : Compare, gaussian function, probability density, energy stored in a spring and Equipartition Theorem. Leads to: E.-L. Florin, A. Pralle, E.H.K. Stelzer, J.K.H. Hörber, Photonic force microscope calibration by thermal noise analysis Applied Physics A 66, S75-S78 (1998)

Histogram of Bead Positions 2 b. Position Sensitive Detector voltages corresponding to bead positions in trap.

Histogram of Bead Positions 3 c. Histogram of centralized data and fitting to a gaussian

Histogram of Bead Positions 4 d. Function to fit, parameter values, trap stiffness Units are important !!!

Power Spectrum of Bead Positions 1 a. Motivation and preliminary calculations 1. Idea : If the motion of a bead in trap obey the Langevin equation, the solution of differential equation can be utilized to find stiffness !!! Application : In the absolute value square of fourier transform of differential equation (power spectrum), corner frequency is related with stiffness. Math : Start with the Langevin equation, add a time dependent noise, find the fourier transform, calculate magnitude of fourier transform. Langevin Equation :

Power Spectrum of Bead Positions 1 a. Motivation and preliminary calculations 2. Ignore inertial terms (Low Reynolds number), take the force term as spring force Also take the noise as Brownian noise. Then… with& Take the fourier transform of equation gives… Magnitude square of the fourier transform of positions (power spectrum) is… with Joshua W. Shaevitz, A Practical Guide to Optical Trapping, ??? (2006)

Power Spectrum of Bead Positions 1 a. Motivation and preliminary calculations 3. Having only PSD voltage values corresponding to actual position values Leads us to change equation for power spectrum correspondingly… with G. Romano, L. Sacconi, M. Capitanio, F.S. Pavone, Force and Torque Measurements using magnetic micro beads for Single molecule biopysics, Optics Communications 215, (2003)

Power Spectrum of Bead Positions 2 b. Power Spectrum of centralized data and fitting to a lorenzian…

Power Spectrum of Bead Positions 3 c. Function to fit, parameter values, trap stiffness 1 With…

Power Spectrum of Bead Positions 3 c. Function to fit, parameter values, trap stiffness 2 Units are important !!! Using the scaling factor for detector, stiffness coefficients are…

Stokes Flow or Stokes Friction 1 a. Motivation and preliminary calculations 1. Idea : If we move our trap with constant velocity while we there is a bead in it, equilibrium position of the trapped bead should change !!! Application : The new equilibrium position caused from constant velocity should be related with stiffness of the trap !!! Math : Start with the Langevin equation, add a constant velocity term, rearrange terms to observe shift in equilibrium position. with

Stokes Flow or Stokes Friction 1 a. Motivation and preliminary calculations 2. Approximated stage position in time v = constant !!! Approximated force on bead in time hence the equilibrium position

Stokes Flow or Stokes Friction 2 b. Centralized bead position voltages …

Stokes Flow or Stokes Friction 2 b. Histogram of centralized data and fitting to a double gaussian…

Stokes Flow or Stokes Friction 3 c. Fitting function, parameter values, trap stiffness 1 Now, we utilize the scaling factor belongs detector (from power spectrum calculation) and another scaling factor belongs stage (from company manual) for calculating xeq and v.

Stokes Flow or Stokes Friction 3 c. Fitting function, parameter values, trap stiffness 2 Using the relation

Results & Discussion Calculated trap stiffness values are; Stiffness values are similar to the values are calculated in article below … They are (same order of magnitude) close but not so close to tell a certain stiffness value... The scale factor for detector may need a check for its accuracy. This is postponed to another presentation… Fitting the histogram of positions to double gaussian gives similar width values so that similar stiffness values. But there are almost 20% difference among them… If the response of the detector to a single axis movement on the stage is a function of all four detector voltages, all of above analysis will need refinement… If the azimuthal symmetry of the incoming gaussian beam somehow disturbed, again all of analysis will need refinement… L. Oddershede, S. Grego, S.F. Norrelykke, K. Berg-Sorensen, Optical Tweezers: Probing Biological Surfaces Probe Microscopy 2, (2000)