Optical tweezers and trap stiffness

Dept. of MechatronicsYong-Gu Lee

1. Bechhoefer J and Wilson S. 2002. Faster, cheaper, safer optical tweezers for the undergraduate laboratory. Am. J. Phys. 70: 393-400.

Potential well of laser trapping

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.

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.

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.

2

23

1 1

2 2

1.38 10 [Joule/Kelvin]

B

B

E k T k x

k

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

1kt

b tx x e

k

k

Trap stiffness calculation by power spectrum

Ref. 1. Bechhoefer J and Wilson S. 2002. Faster, cheaper, safer

optical tweezers for the undergraduate laboratory. Am. J. Phys. 70: 393-400.

2. Rolf Rysdyk, Automatic Control of Flight Vehicles class notes: stochastic signal models, http://www.aa.washington.edu/research/afsl/coursework/aa518/aa518.shtml

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.

2

221

2

x

pdf x e

2

2

2 2 20 ( )

N

i

x

xx pdf x x

N

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

is the deviation from the equilibrium

is the particle mass

is the friction coefficient

is the trap spring constant

is the fluctuating force due to random

kicks by the many

mx x kx t

x

m

k

t

neighboring fluid molecules

, the friction coefficient• 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

6 R [dimension, kg/s]

Viscosity [dimension, Stress* Time, ]

2 2

1kg m kgs

s m s m

max max/ 6 , where is the maximum force exerted by the tweezers.F a F

Assume overdamped motion

0 0 0

00

1 1, is the relaxation time.

Solving for ,

1t

mx x kx t

x x t k

x

x e e d Ce

Ref. Paul D. Ritger and Nicholas J. Rose, Differential equations with applications, McGraw-Hill, 1968, pp. 40-41

:general solution

Autocorrelation• The correlation of a signal y(t) with itself is referred to as

autocorrelation. Consider there is a time delay during the correlation,

,

,

,

if and , are independent to each other

,

i i

i j

i j

i

i i i i

i i

i j i j

i j i j

i i

f pdf

pdf

pdf pdf

pdf

,

2 2

0

if

,

j

i j

i

j j

i i i

i j i j

i i

pdf

pdf

pdf

Autocorrelation function

• Autocorrelation function x t x t

0

0 0

00

00

0

0

2

0

2

0

1

ex x e d e d

e d Ce

ee d C C e

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

Autocorrelation function is then obtained by taking an ensemble average, bearing in mind that the only

stochastic (random) term is

0

0

00

00

0

0

2

0

2

0

1

ex x e d d

e d Ce

ee d C C e

• Since successive random kicks are uncorrelated with each other

M

Equipartition theorem

0

00

00 0

0 0

0

0 22

20 2 2

2

0

2

= 2

ex x M e d d C e

e MMe d C e e C e

k

Mek

2

Energy of a single particle :

1 1 1

2 2 2 2

2

B

B

E

ME k T k x k

k

M k T

The final form of the autocorrelation function and

power spectral density

0

20

12 1 12 =

1 22

B B Bk T k T k T kx t x t e e e

k k k

2

2 02 2 2 2 2

2 2 02

0

2

2 2 2 20

2 22 2 22 2 2

2 2 2 2 2 20

1

2 2 11 1 44

142 2 141 2

4 4 4

B B

B B B

k T k TX

k k

kk T k T k T

k kk k

which gives the characteristic frequency response of the particle to the thermal driving force, normalized so that

Using Wiender-Khintchine theorem, one can calculate the power spectrum of the particle by taking the Fourier transform of above to find

2 220

2 1

4 2 2B Bk T k Tk

X d xk k

0

1

2

2 24c

kf

Corner frequency

Wiener-Khinchin Theorem

22

22

2 2

2

2

if tift

if tift

i f f t if

if

if

x t x t X f e df X f e df dt

X f e df X f e df dt

e dt e X f X f df df

f f X f X f e df df

X f f f X f e df

2

2 2

if

if

df

X f X f e df

X f e df

0

2

2

2

20

if t

if t

if t

ift

X f x t e dt

x t X f e dt

e df t

F h t t H f e

Ref. Weisstein, Eric W. "Wiener-Khinchin Theorem." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html

x t x t

Fourier transform of the autocorelation

2

2 2 2

2

2

2

if

if if

x t x t e d

X f e e d df

X f f f df

X f

X

Some example plots

In[15]:= Plot 10.54 x2 0.52

,x,0,2;

0.5 1 1.5 2

2.5

5

7.5

10

12.5

15

In[11]:= Plot124 x2 22

,x,0,2;

0.5 1 1.5 2

0.035

0.045

0.05

0.055

0.06

Measurement from experiments

Power spectrum obtained for a 1 micrometer big polystyrene bead in a trap.Taken from http://www.nbi.dk/~tweezer/introduction.htm

Arbitrary units is sufficient

Comparison of methodsViscosity Geometr

yCCD Thermom

eterPiezo Stage QPD Remarks

Escape force method

T T T Nonlinear region

Drag force method

T T T T Nonlinear region

Equipartition method

T T Linear region, Need to know Volt,displacement relations (calibration is necessary)

Power spectrum method

T T T T Linear region, No calibration necessary.

Step response method

T T T T Linear region