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Department of Physics
Universidade Federal de
Minas Gerais
Belo Horizonte - MG - Brazil
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Belo Horizonte
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I Summer School on Optics and Photonics
19-22 January of 2010, Concepción, Chile
Oscar N. Mesquita
Departamento de Física, ICEX, Universidade Federal de Minas Gerais
Belo Horizonte, Brasil
Prof. Ubirajara Agero
Prof. Márcio S. Rocha (UFV)
Edgar Casas (post-doc)
Dr. Giuseppe Glionna
Lívia Siman (Doctorate)
Ulisses Andrade (Master)
Colaborators
Prof. Moysés Nussenzveig (UFRJ)
Prof. Nathan Bessa Viana (UFRJ)
Profa. Lucila Cescato (Unicamp)
Profa. Simone Alexandre (UFMG)
Profa. Aline Lúcio (UFL)
Prof. Paulo Américo Maia Neto (UFRJ)
Prof. Carlos Henrique Monken (UFMG)
Prof. Ricardo Gazzinelli (UFMG)
Prof. Ricardo Wagner Nunez (UFMG)
Sponsors
Fapemig, CNPq, Finep, Instituto do Milênio de Nanotecnologia, Instituto do Milênio de Óptica Não-linear, Fotônica
e Biofotônica e Instituto Nacional de Fluidos Complexos
Lecture 1
Optical tweezers: basic concepts and comparison between experiments and an
absolute theory
This lecture will be largely based on the articles by A. Ashkin, by Mazolli, Maia Neto
and Nussenzveig and on the doctorate thesis of Alexander Mazolli (UFRJ, 2003) and doctorate
thesis of Márcio Santos Rocha (UFMG, 2008).
Lecture 2
Application of optical tweezers in single-molecule experiments with DNA
This lecture will be based on our own work with additional examples from other laboratories worldwide.
Lecture 3
Defocusing Microscopy: a new way of phase retrieval and 3D imaging of
transparent objects
Defocusing Microscopy (DM) is a technique developed in our laboratory for full-field phase retrieval
and 3D-imaging of transparent objects, with applications in living cells.
Lecture 4
Application of defocusing microscopy to study living cell motility
We apply DM to study motility of macrophages and red blood cells. Some recent theoretical
elasticity models for the coupling between cytoskeleton and lipid bilayer will be discussed.
Lecture 1
Optical tweezers: basic concepts and comparison between experiments and an
absolute theory
Schematic set-up of optical tweezers
Optical Tweezers is an invention of A. Ashkin in 1970
A. Ashkin, Acceleration and trapping of particles by radiation pressure, Phys. Rev. Lett. 24, 156 (1970)
Competition between gradient force and force due to radiation pressure
The gradient force must overcome the force due to radiation pressure for optical trapping
Optical tweezers experiments were the precursor of trapping of atoms with lasers
A. Ashkin, Trapping of atoms by resonance radiation pressure, Phys. Rev. Lett. 40, 729 (1978)
Ashkin also reported experiments of optical trapping of cells and other biological material
A. Ashkin, J. M. Dziedzic, Optical trapping and manipulation of virus and bacteria, Science 235, 1517
(1987)
A. Ashkin, J. M. Dziedzic, Optical trapping and manipulation of single cells using infra-red laser traps,
Phys. Chem 93, 254 (1989)
Qualitative ideas on gradient and radiation pressure forces
Light carries momentum
Momentum
conservation
Geometric optics description (l<<a)
Figures below are from Mazolli’s thesis
Cilindrical Beam – refracted (gradient force)
Centered (refracted)
Out-of-center (refracted)
Cilindrical Beam – reflected (radiation
pressure)
Centered (reflected)
Out-of-center (reflected)
Conical Beam – refracted (gradient force)
Centered (refracted)
(focus above the sphere center)
Centered (refracted)
(focus below the sphere center)
Out-of-center (refracted)
(focus above the sphere center)
Conical Beam – reflected (radiation pressure)
Rayleigh limit description (l>>a)
Electric dipole in an inhomogeneous electric field
Only the gradient force exists in this limit
Where
a is the particle radius
Consequently the stiffness K is
proportional to a 3
Order of magnitude estimate of the gradient force in the geometric optics
(GO) regime for one ray refracting through a glass sphere in water with T~1.
where nH = 1.33 and nV = 1.50
are the index of refraction of
the water and glass. For Pot =
1 mW and a = 45o, the
gradient force is Fg = 0.95 pN.
For small displacements, sina
~a and sinb~b, with a ~ ( x / a ).
Then
Pot  nH
1 
FR  2nH
c  nv
 x 
 
 a 
and
1
Ka  
a
Important scalings
Geometric optics limit
l  a
In the geometric optics regime the magnitude of the force will be a function of the
displacement of the sphere from the equilibrium position divided by its radius:
Fx  F ( x / a)
For small displacements in relation to the equilibrium position
 x
Fx a  
a
and
1
Ka  
a
Rayleigh limit
l  a
K a a3
In these earlier calculations of optical forces on particles the incident beam from a high
NA objective was not properly described. Even in the GO limit, although the proper
scaling was obtained, the correct value for the gradient force was not obtained.
Basic ingredients for modeling optical tweezers forces – Mie-Debye (MD) theory
-Proper description of the highly focused laser beam which comes out from a larger
numerical aperture objective.
-Since a complete theory has to be valid from the Rayleigh
limit up to the geometric optics limit, Mie theory has
to be used in order to have a description valid for any bead size.
-Both requirements were only recently accomplished with the
complete theory of optical tweezers for dielectric spheres by Maia Neto and
Nussenzveig (Europhys. Lett, 50, 702 (2000)), and Mazolli, Maia Neto and Nussenzveig
(Proc. R. Soc. Lond. A 459, 3021 (2003)), named Mie-Debye (MD) theory.
-Solving the problem for trapped spheres is important, because spheres can be used as
handles in several applications, where forces in the pN range ought to be exerted.
Mazolli, Maia Neto, Nussenzveig theory of optical tweezers (MD theory)
Modeling the incident beam from a high NA objective
1) Abbe sine condition
2) Richards-Wolf approach
Gaussian laser incident field
Abbe sine condition for objectives (minimum aberration):
objective
with
Then the electric field Eout is proportional to
sine condition. The fields are then:
implies
as a result of the Abbe
Incident beam
before the objective
After the objective
Electric and magnetic fields can be derived from the Debye potentials below
Where d M , M  are the matrix elements o finite rotations and JM are Bessel functions of
integer order. Once E and H have been obtained from the Debye potentials, the Maxwell
stress-tensor can be calculated and finally the total force on the sphere can be
determined. There is no doubt that this problem is a “tour of force” on electromagnetic
theory.
j
Relation between Debye potentials and the fields
Total fields: internal plus
external
Maxwell stress-tensor
Force on the sphere
Results
Measurement of the local power at the focus of a high numerical
aperture objective
Microbolometer
Viana, Mesquita & Mazolli, APL 81, 1765 (2002)
The microbolometer consists of small droplets in the micron size of Hg in water. We
shine one of this droplet with the laser, which we want to measure the intensity at the
focus of the objective. The laser beam heats the Hg droplet. The temperature at the
surface of the droplet achieves steady-state in a fraction of second. As one slowly
increases the laser power, the droplet heats up, until it achieves the water boiling
temperature and then jumps. This jump is very easy to detect.
A=0.272 for l=832nm
PL=A.Pa
T0 is the laboratory temperature;
T is the boiling temperature of water
when the bead jumps;
R is the radius of the Hg droplet;
Pa is the absorbed power;
PL is the local power we want;
A is the absorption coefficient of Hg.
Measurement of the beam profile entering the objective
Mirror method
Take the objective and replace it
by a mirror .
Dual objective method
Standard method to measure
the local power at the focus
of high numerical aperture
objetives. One has to be
careful because the transmission
coefficients of objectives in the
IR are not spatially uniform,
and changes the beam profile, as
shown by Viana, Rocha,
Mesquita, Mazolli, and Maia
Neto, Appl. Opt. 45, 4263 (2006)
Measurements of stiffness using oil droplets trapped by an optical tweezers
/P
L
(pN/m mW)
4
3
2
1
0
-1
0
0.5
1
1.5
a (m)
2
2.5
3
The discrepancy between theory and experiment suggests that the inclusion of spherical
aberrations into the theory is important. This has been done and received the name MieDebye-Spherical-Aberration (MDSA) theory. It is important to mention that, since in the
theory there are no adjustable parameters, all parameters used have to be measured:
bead radius, refractive indices of the bead and medium, profile of the incident beam
(filling factor), and the local laser power at the objective focus. The experimental
procedures used will be discussed in the next lecture.
The spherical aberration between the glass slide and the medium tends to deteriorate
the performance of optical tweezers: as much is the bead trapped away from the glassslide worse becomes the optical tweezers.
Spherical aberration effects - (MDSA) theory
Results
Comparison between MDSA theory and
experiment. Effects of limited objective
filling factor, and spherical aberration
clearly appearing.
Multiple minima due to spherical aberration
Viana, Rocha, Mesquita, Mazolli,
Maia Neto and Nussenzveig, PRE (2007).
Experimental implementation of optical tweezers
Schematic set-up
Set – up at
UFMG
Brownian motion of a microsphere in a
harmonic potential
Langevin equation:
d 2x
dx
m 2 
 kx  f (t )
dt
dt
 f (t ) f (t )  2k BT (t  t )
Position correlation function satisfies the equation:
d 2  x(0) x(t ) 
d  x(0) x(t ) 
m

 k  x(0) x(t )  0
2
dt
dt
Neglecting inertia and using the equipartition theorem
k BT
 x 
ki
2
k BT
xi 0xi t  
e
ki
i 

ki

6a
ki

t
i
Back-scattering profile
One moves the trapped bead in relation to the probe He-Ne laser
Back-scattering profile from a polystirene bead with the same diameter, 2.7 m  5% ,
as in the previous slide.
Crosses are the backscattering
profile with the detector in the
center.
Losanges are the backscattering
profile with the detector moved
to maximize the intensity of
one of the lateral peaks.
As compared to the previous
slide, the central peak now
has minimum intensity. This
effect and the lateral peaks can
be explained by the
MD theory.
Mazolli, Maia Neto and Nussenzveig - MD theory
(m)
Backscattering intensity of droplets of CCl in water.
4
The oscilations are due to interference as the size
of the droplets changes in time.
100
This effect has potential
application in colloidal
physics,
for determination of
refractive index of coloidal
particles and studies of
coloidal growth.
80
I (kHz)
60
40
20
0
0
20
40
60
80
time (s)
100
120
140
160
Calibration
 
 f x
Backscattering profile which can be fit with a function I x  I 0 e
,
where f(x) is a polynomium. Then we have an expression that relates I (the scattered
intensity) and position (x) of the center of mass of the microspheres.
I xi   I 0 e
 f  xi 
 dI 
I ( xi  x0i )  I ( x0i )    ( xi  x0i )
 dxi  x0 i
 f 
a i   
 xi 
 I (0) I (t ) 
2
2




g t  

1

a

x
0

x
t

a
x
z z 0 z t 
2
 I ( x0i ) 
( 2)
2
xi  ( xi  x0i )
Here we are assuming that motion in x and y are equivalent, which is the case if the
incident beam on the sphere has radial simmetry.
For an expansion around xi = 0
 dI  then,
0
 
 dxi  x0 i 0
 d 2 I  ( xi  x0i ) 2
I ( xi  x0i )  I ( x0i )   2 
2
 dxi  x0 i
 I (0) I (t ) 
2
2
2
2
2
2






t 
g t  

1

b

x
0

x
t

b

z
0

z
x
z
2
 I ( x0i ) 
( 2)
2
xi  ( xi  x0i )
In this case the intensity correlation function is related to the second order correlation of
bead center of mass position.
Trapped bead oscillating with a fixed frequency along the x-direction
Note that g(2)(t) for the bead
located at x0 = 0 in the
backscattering profile has twice
the frequency of g(2)(t) for the
bead at x0 > 0.
First (<x(0)x(t)>) and second order (<x2(0)x2(t)>) correlation functions,
depending on the position of the bead in the scattering profile.
Correlation function obtained in the linear part of the scattering
profile, where clearly two time constants appear: a shorter one for motion
perpendicular and the longer one parallel to the incident direction.
Results
k x  0.0058  0.0002 dyn / cm
From the time constant
k x  0.0059  0.0003 dyn / cm
From <x2>
Since from the measurements one can get the stiffness k and the
friction coefficient , one can check how the friction changes as
the bead approaches the glass slide. One can move the bead in relation to the glass
slide by just moving the objective.
Parallel Stokes friction near a wall (Faxen’s expression)
 //  9  a  1  a 
45  a 
1 a
 1       
    
 0  16  h  8  h  256  h  16  h 
3
4
where a is the radius of the bead, h is the distance from its centerof-mass to the glass slide, and
.
 0  6a
5 1



Viana, Teixeira, and
Mesquita, PRE (2002)
 0  2.27 10 5 g / s
which agrees within 5% with the expected value for
this bead in water.
Summary
Trapping of a dielectric particle by a laser is a competition between radiation pressure
(due to reflection) and gradient forces (due to refraction).
The exact theory MDSA is the most complete theory of optical tweezers.Our data are in
support of the theory.
We measure the stiffness of our optical tweezers (polystirene bead of 3m trapped by
an Infra-red laser), via Brownian fluctuations of the trapped bead. These fluctuations are
probed via back-scattering of a He-Ne laser.
By obtaining the time correlation function of the bead position fluctuations, we accurately
measure the stiffness of the the optical tweezers.