1. Optical tweezers and trap stiffness
Dept. of Mechatronics
Yong-Gu Lee
Bechhoefer J and Wilson S Faster, cheaper, safer optical tweezers for the undergraduate laboratory. Am. J. Phys. 70:

2. Potential well of laser trapping

3. Escape force method
This method determines the minimal force required to pull an object free of the trap entirely, generally accomplished by imposing a viscous drag force whose magnitude can be computed
To produce the necessary force, the particle may either be pulled through the fluid (by moving the trap relative to a stationary stage), or more conventionally, the fluid can be moved past the particle (by moving the stage relative to a stationary trap).
The particle velocity immediately after escape is measured from the video record, which permits an estimate of the escape force, provided that the viscous drag coefficient of the particle is known. While somewhat crude, this technique permits calibration of force to within about 10%.
Note that escape forces are determined by optical properties at the very edges of the trap, where the restoring force is no longer a linear function of the displacement. Since the measurement is not at the center of the trap, the trap stiffness cannot be ascertained.
Escape forces are generally somewhat different in the x,y,z directions, so the exact escape path must be determined for precise measurements. This calibration method does not require a position detector with nanometer resolution.

4. Drag Force Method
By applying a known viscous drag force, F, and measuring the displacement produced from the trap center, x, the stiffness k follows from k=F/x.
In practice, drag forces are usually produced by periodic movement of the microscope stage while holding the particle in a fixed trap: either triangle waves of displacement (corresponding to a square wave of force) or sine waves of displacement (corresponding to cosine waves of force) work well
Once trap stiffness is determined, optical forces can be computed from knowledge of the particle position relative to the trap center, provided that measurements are made within the linear (Hookeian) region of the trap. Apart from the need for a well-calibrated piezo stage and position detector, the viscous drag on the particle must be known.

5. Equipartition Method
One of the simplest and most straightforward ways of determining trap stiffness is to measure the thermal fluctuations in position of a trapped particle. The stiffness of the tweezers is then computed from the Equipartition theorem for a particle bound in a harmonic potential:
The chief advantage of this method is that knowledge of the viscous drag coefficient is not required (and therefore of the particle’s geometry as well as the fluid viscosity). A fast, wellcalibrated position detector is essential, precluding video-based schemes.

6. Step Response Method
The trap stiffness may also be determined by finding the response of a particle to a rapid, stepwise movement of the trap
For small steps of the trap, xt the response xb is given by below where is trap stiffness and is the viscous drag.
harder to identify extraneous sources of noise or artifact using this approach. The time constant for movement of the trap must be faster than the characteristic damping time of the particle

7. Trap stiffness calculation by power spectrum
Ref.
Bechhoefer J and Wilson S Faster, cheaper, safer optical tweezers for the undergraduate laboratory. Am. J. Phys. 70:
Rolf Rysdyk, Automatic Control of Flight Vehicles class notes: stochastic signal models,

8. Normal probability distribution
If the values of x are normally distributed about a zero mean, then ‘histogram’ distribution x would look like:
where is the variance and N is the number of samplings that is sufficiently large.

9. A particle immersed in a fluid and trapped in a harmonic potential undergoes following dynamics

10. For scanning optical tweezers the velocity of the scan must satisfy
If the particle is a sphere of radius R immersed in a fluid of viscosity and far from any boundaries, then standard hydrodynamic arguments lead to
For scanning optical tweezers the velocity of the scan must satisfy
[dimension, kg/s]
Viscosity [dimension, Stress* Time, ]

11. Assume overdamped motion
They are characterized by drag or friction forces large compared to the inertial forces of the system.
gamma > 2m sqrt(k/m)
:general solution
Ref. Paul D. Ritger and Nicholas J. Rose, Differential equations with applications, McGraw-Hill, 1968, pp

12. Autocorrelation
The correlation of a signal y(t) with itself is referred to as autocorrelation. Consider there is a time delay during the correlation,

13. Autocorrelation function
Note we have replaced t by 0, this is allowed because we are assuming that we are assuming that x(t) is a stationary stochastic process, so that ensemble averages are independent of the time at which they are carried out

14. Autocorrelation function is then obtained by taking an ensemble average, bearing in mind that the only stochastic (random) term is

15. Since successive random kicks are uncorrelated with each other

16. Equipartition theorem
From the equipartition theorem we know that every degree of freedom has energy kBT/2 where kB is the Boltzmann constant and T is the temperature of the medium in Kelvin.

17. The final form of the autocorrelation function and power spectral density
Using Wiender-Khintchine theorem, one can calculate the power spectrum of the particle by taking the Fourier transform of above to find
Corner frequency
The Wiener–Khinchin theorem (also known as the Wiener–Khintchine theorem and sometimes as the Khinchin–Kolmogorov theorem) states that the power spectral density of a wide-sense-stationary random process is the Fourier Transform of the corresponding autocorrelation function. Ref.
Corner frequency is frequency when integrated from 0 gives half of the total energy
which gives the characteristic frequency response of the particle to the thermal driving force, normalized so that

18. Wiener-Khinchin Theorem
Ref. Weisstein, Eric W. "Wiener-Khinchin Theorem." From MathWorld—
A Wolfram Web Resource.

19. Fourier transform of the autocorelation

20. Some example plots

21. Measurement from experiments
Arbitrary units is sufficient
The difference in decibels between the two is defined to be
10 log (P2/P1) dB        where the log is to base 10.
The Corner Frequency is the Frequency of the Half Power Point (or -3dB Point),.
Power spectrum obtained for a 1 micrometer big polystyrene bead in a trap.
Taken from

22. Comparison of methods Viscosity Geometry CCD Thermometer Piezo Stage
QPD
Remarks
Escape force method T
Nonlinear region
Drag force method
Equipartition method
Linear region, Need to know Volt,displacement relations (calibration is necessary)
Power spectrum method
Linear region, No calibration necessary.
Step response method
Linear region