Lecture 8: Measurement of
Nanoscale forces II
What did we cover in the last
lecture?
Atomic force microscopes can be
used to measure forces
F
The spring constant of an AFM
cantilever is determined by its
material properties and its physical
dimensions
A split photodiode arrangement is
used to detect deflections and to
measure forces with >10pN
precision
F  kz
3EI
k 3
l
In this lecture…
Optical tweezers
The wave nature of light
Optical trapping
A point dipole in a field
revisited
Using lasers to trap particles
Applications of optical trapping
Optical tweezers
Optical tweezers provide
a way of manipulating
nano or micron scale
objects and measuring
very small forces.
Particles are usually
trapped by focussing
visible light (usually a
laser)
Objects can be
manipulated by moving
the laser beam
Visible
light
Particle
The wave nature of light
Light is an electromagnetic wave.
As it propagates through vacuum
or a medium it carries energy.
Energy is stored in electric and magnetic
fields that oscillate in a direction
perpendicular to the direction of travel
(more on this in Classical Fields)
E  Eo sin( t  kx)
Electric field
travelling in
positive x direction
The intensity of light
The time averaged intensity of light, I, can be related to the
magnitude of the electric field, E, by the relationship
c o n 2
I 
E
2
in Wm-2
Where c is the speed of light (ms-1) and n is the refractive
index of the medium in which it propagates
Note: For a sinusoidal oscillating field, the time average
value of <E2> is not zero. So the intensity is not zero!
Optical trapping
Light Beam
Small dielectric particles
experience a force when
illuminated with visible light.
If a dielectric particle is placed
in an intensity gradient a force
is exerted on the particle such
that it will move to a region of
higher intensity
Intensity gradient
A point dipole in a field
Suppose we treat the small particle as a collection of small
dipoles
The energy associated with placing the particle in an
electric field E is
U   p.E
~
Where p is the electric
dipole moment (see
lecture 3)
~
For a particle of polarisability, , we obtain the energy as
U   E . E   E
~
~
~
2
We can use this to
calculate the force on
the particle (see OHP)
The force on a dipole in an intensity
gradient
In one dimension, the force on a dielectric particle is
dU
2 dI
F 
i
i
dx ~ co n dx ~
Generalising to 3 dimensions we have
dU
dU
dU
2  dI
dI
dI 
 i 
F 
i
j
k
j  k 
dx ~ dy ~ dz ~ co n  dx ~ dy ~ dz ~ 
So the force always acts along the direction of increasing
intensity gradient. Particles move to regions of higher
intensity
Creating a gradient in intensity
Laser Beam
wo
The beam profile of a laser is not
uniform - it has a Gaussian profile in
the radial direction – trapping in x-y
plane only
 (x2  y2 ) 

I  I o exp  
2
2 wo 

 r2 
 I o exp   2 
 2 wo 
wo is a measure of the beam width
However, there is a problem! We
only get trapping in x-y plane – no
gradient in z direction!
Optical Trapping in 3D
Microscope
objective

Ambient
refractive
index, n
We can use a microscope objective
to create a 3D trap!
The changing area of the beam near
the focus means that the intensity of
the light decreases on either side of
the focal point
This traps the particle in z direction
also!
The maximum gradient in intensity
can be obtained by using a lens with
a high numerical aperture (NA)
NA  n sin 
Gaussian beam profile: quadratic
approximation (trap stiffness)
For small displacements from the central position we can
approximate the Gaussian using a quadratic intensity
profile
Laser Intensity
I  Io
1  bx 
2
Intensity gradient
dI
 2 I o bx
dx
Problem 1
The intensity profile of a laser beam is given by the equation
 r2 
I  I o exp   2 
 2wo 
Where wo is the radius of the beam and Io is the intensity at
its centre.
Verify that for small displacements (r<<wo) the intensity
profile can be approximated by a quadratic form. Hence
show that the force exerted on a small particle at the centre
of the beam can be expressed in the form F=-kr
Derive an expression for k.
Measuring forces with optical
tweezers
As we saw in the previous problem, for small
displacements, x, the force on a particle in a focussed laser
beam is given by
F  kx
Where k is the trap stiffness and is given by
I o
k
co nwo2
So if we can measure the displacement we can determine
the force
Detecting deflections: Quadrant
photodiode
The split photodiode arrangement used in AFM can also be
used to detect deflections in optical traps. However a
quadrant photodiode is used to detect deflections in both
the x and y directions
Shadowing (large particles)
Fringe pattern (small particles)
Laser beam
Quadrant
photodiode
(4 photodiodes)
Trap calibration
If we want to calibrate
a trap we can measure
the displacement of a
particle under the
influence of known
forces
Fapplied
Ftrap
x
Or we can measure the
rms displacement of a
trapped particle under the
influence of thermal
motion
k
Fapplied
x
k BT
k~
2
x 
Problem II
A 100nm radius polymer particle having a relative
permittivity of p=5 is suspended in water w=80, nw=1.33.
A 10W laser beam with a Gaussian profile is focussed
down on to the particle such that it creates an optical trap
with an effective radius of 1 mm.
Calculate the displacement of the trapped particle if a force
of 1pN is applied to it.
Multiple traps: time sharing
If traps are switched on and off faster than particles can
diffuse away, the same trap can be moved around and used
to trap many objects
A game of Tetris
with glass beads!
http://www.nat.vu.nl/~joost/tetris/
Force Measurements
It is also possible to perform more serious measurements
on e.g. single molecules
An optical trap and a fine
micropipette can be used
to measure the forces
exerted by individual
DNA molecules
Small polymer beads are
tethered to the ends of
the molecule
C. Bustamante et al.,
Nature, 421(23), 423 (2003)
Summary of key concepts
Particles in a laser beam will experience a
force that acts to pull them to regions of
higher intensity
F 
dU
2 dI
i
i
~
~
dx
co n dx
Lasers have a Gaussian beam profile which
naturally lends itself to trapping of particles
Trapping in 3 dimensions can be achieved by
focussing the laser using a microscope objective
Trapping forces on nanoscale particles can be measured
with pN precision