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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)
Prof. Moysés Nussenzveig (UFRJ) Prof. Paulo Américo Maia Neto (UFRJ)
Prof. Nathan Bessa Viana (UFRJ) Prof. Carlos Henrique Monken (UFMG)
Profa. Lucila Cescato (Unicamp) Prof. Ricardo Gazzinelli (UFMG)
Profa. Simone Alexandre (UFMG) Prof. Ricardo Wagner Nunez (UFMG)
Profa. Aline Lúcio (UFL)
Fapemig, CNPq, Finep, Instituto do Milênio de Nanotecnologia, Instituto do Milênio de Óptica Não-linear, Fotônica e Biofotônica e Instituto Nacional de Fluidos Complexos
Optical tweezers: basic concepts and comparison between experiments and an
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).
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 world-wide.
Defocusing Microscopy: a new way of phase retrieval and 3D imaging of
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.
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.
Optical tweezers: basic concepts and comparison between experiments and an
Schematic set-up of optical tweezers
A. Ashkin, Acceleration and trapping of particles by radiation pressure, Phys. Rev. Lett. 24, 156 (1970)
Geometric optics description (l<<a)
Figures below are from Mazolli’s thesis
Cilindrical Beam – refracted (gradient force)
Cilindrical Beam – reflected (radiation pressure)
Conical Beam – refracted (gradient force)
(focus above the sphere center)
(focus below the sphere center)
(focus above the sphere center)
Rayleigh limit description (l>>a)
Electric dipole in an inhomogeneous electric field
Only the gradient force exists in this limit
Where is the particle radius
Consequently the stiffness K is
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 ~ .
Geometric optics limit
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:
For small displacements in relation to the equilibrium position
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.
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):
Then the electric field Eout is proportional to as a result of the Abbe sine condition. The fields are then:
before the objective
After the objective
Electric and magnetic fields can be derived from the Debye potentials below
Where 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.
Total fields: internal plus external
Force on the sphere
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
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.
Take the objective and replace it
by a mirror .
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
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 Mie-Debye-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 glass-slide worse becomes the optical tweezers.
Spherical aberration effects - (MDSA) theory
Comparison between MDSA theory and
experiment. Effects of limited objective
filling factor, and spherical aberration
Viana, Rocha, Mesquita, Mazolli,
Maia Neto and Nussenzveig, PRE (2007).
Position correlation function satisfies the equation:
Neglecting inertia and using the equipartition theorem
One moves the trapped bead in relation to the probe He-Ne laser
Back-scattering profile from a polystirene bead with the same diameter, , as in the previous slide.
Crosses are the backscattering
profile with the detector in the
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
for determination of refractive index of coloidal particles and studies of coloidal growth.
Backscattering profile which can be fit with a function ,
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.
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.
In this case the intensity correlation function is related to the second order correlation of bead center of mass position.
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.
profile, where clearly two time constants appear: a shorter one for motion perpendicular and the longer one parallel to the incident direction.
friction coefficient g, 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)
where is the radius of the bead, h is the distance from its center-
of-mass to the glass slide, and .
which agrees within 5% with the expected value for this bead in water.
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 3mm 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.