Optical tweezers and trap
stiffness
Dept. of Mechatronics
Yong-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
1
1
harmonic potential:
E  k BT  k x 2
2
2
k B  1.38 1023 [Joule/Kelvin]
• 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.
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 k is trap stiffness and  is the
kt
viscous drag.
 

xb  xt 1  e





• 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 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/aa
518/aa518.shtml
Normal probability distribution
• If the values of x are normally distributed
about a zero mean, then ‘histogram’
distribution x would look like:
x2
 2
1
pdf  x  
e 2
2
2
where  is the variance and N is the number
of samplings that is sufficiently large.
N
2 
2
x

i 0
N
  x 2 pdf ( x)  x 2
x
A particle immersed in a fluid and
trapped in a harmonic potential
undergoes following dynamics
mx   x  kx    t 
x is the deviation from the equilibrium
m is the particle mass
 is the friction coefficient
k is the trap spring constant
  t  is the fluctuating force due to random
kicks by the many 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
  6 R
[dimension, kg/s]
• For scanning optical tweezers the
velocity of the scan must satisfy

Fmax /  6a  , where Fmax is the maximum force exerted by the tweezers.
Viscosity [dimension, Stress* Time,
kg  m 1
kg
 2 s 
s2
m
sm
]
Assume overdamped motion
mx   x  kx    t 
x
1
0
x
  t  , 0   k is the relaxation time.

1
Solving for x,
x    e

0

t

1

    e
 
0
d   Ce

0
:general solution
Ref. Paul D. Ritger and Nicholas J. Rose, Differential equations with applications, McGraw-Hill, 1968, pp. 40-41
Autocorrelation
• The correlation of a signal y(t) with itself is referred to as
autocorrelation. Consider there is a time delay during the
correlation,
         f       

   
        
  i   i pdf   i ,   i
  i ,  i
if  i   i and     ,     are independent to each other










   
           pdf      pdf     
   

  i   j pdf   i ,   j
  i ,  j
i
j
i
j
  i   j

     pdf           pdf       0
i
 
i
  i
j
 
  j 
if  i   i   i










   
     pdf       

  i   j pdf   i ,   j
  i ,  j
2
  i 
2
i
i
j
Autocorrelation function
x t  x t   
• Autocorrelation function

x  0  x   
e
0


    e
0
2
 
0

1
    e


0
 
0



e
0


    e


0

d      e d 

d   Ce

0

0

d   C  C e
2
0
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 

x  0  x   

e
0

0
 
2
1

 
    e d   Ce
0


e

        e
 
0

0





0
  
0
d d 

0
    e d   C  C e
2

0
• Since successive random kicks are
uncorrelated with each other
         M       
Equipartition theorem

x  0  x   
e
0
2
0
 
 


e
0
2


M        e
2 
0

Me  0 d   C 2 e

M  0
=
e
2 k
Energy of a single particle E :
1
1
1 M
2
E  k BT  k x  k
2
2
2 2 k
M   2k BT 

0
   
0
d d   C 2e


M 0
e  C 2e
2 k

0

0
The final form of the
autocorrelation function and
power spectral density
1

kBT  0 k BT 2
2k  T 1  
1 k
 
x t  x t    
e 
e = B2
e
 
k
k 1
k
2
0 
2
Using Wiender-Khintchine theorem, one can calculate the power spectrum of
the particle by taking the Fourier transform of above to find

1
X 2   
 02
2k B T
2k  T
1
 B2
2
1
k
k
1  4 2 2 0 2
2 2

4


2
Corner frequency
0
1

2k B T
k2
k
4  0
2k  T
 B2
1
k
 2
2 2
2
2
4  0
fc 
2
kT
4 
 B2
2
k
2 
 2
2 2
2
4 
2
1
k
2
4 
2
2
k2
4 2 2
 2
which gives the characteristic frequency response of the particle to the
thermal driving force, normalized so that

1
0


0
X 2   d  
2k B  T k
k BT 1 2

 x
2
k
4
2k 2
Wiener-Khinchin Theorem
x t  x t   



2 if  t  
2 ift
    X  f  e df  X  f   e
df  dt
  






2 if  t  
2 ift


df  dt
   X  f  e df  X  f  e


  2 i f  f t  2 if 
    e
dt  e
X  f  X  f   df df 
   

 
 

    f   f  X  f  X  f  e
2 if 
x t  x t   

df df
 





X  f     f   f  X  f  e



 X  f  X  f e





X 2  f  e 2 if  df
2 if 
df
2 if 
df df
Xf 

 x t  e
2 if t
dt

x t  

 X  f e
2 if t
dt


e
2 if t
df    t 

F
h  t  t0   H  f  e 2 ift0
Ref. Weisstein, Eric W. "Wiener-Khinchin Theorem." From MathWorld—
A Wolfram Web Resource. http://mathworld.wolfram.com/Wiener-KhinchinTheorem.html
Fourier transform of the
autocorelation


x  t  x  t    e 2 if  d




2
2 if  2 if 
  X  f   e
e
d  df

 





X 2  f    f  f   df

 X 2   f 
 X 2  
Some example plots
Measurement from experiments
Arbitrary units is sufficient
Power spectrum obtained for a 1 micrometer big polystyrene bead in a trap.
Taken from http://www.nbi.dk/~tweezer/introduction.htm
Comparison of methods
Viscosity
Geometry
CCD
Escape force
method
T
T
T
Drag force
method
T
T
T
Equipartition
method
Power
spectrum
method
T
T
Step response
method
T
T
T
Thermomet
er
Piezo Stage
QPD
Remarks
Nonlinear region
T
Nonlinear region
T
T
Linear region, Need
to know
Volt,displacement
relations (calibration
is necessary)
T
T
Linear region, No
calibration
necessary.
T
Linear region