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### Results potentials below

Localization

Belo Horizonte

Welcome to UFMG

In 1927 UFMG was founded, in order to integrate university and society. It provides space for the diffusion of the

most different cultural expressions. UFMG offers 33 bachelor’s degree programs, 60 master’s degree programs, 70 specialist degree programs, and 53 doctoral programs in our professional colleges and schools.

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. 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)

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

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 world-wide.

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.

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

Lightcarriesmomentum

Momentumconservation

Geometric optics description (l<<a)

Figures below are from Mazolli’s thesis

Cilindrical Beam – refracted (gradient force)

Centered (refracted)

Conical Beam – refracted (gradient force)

Centered (refracted)

(focus above the sphere center)

(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 is the particle radius

Consequently the stiffness K is

proportional to

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 ~ .

Then

and

Important scalings geometric optics (GO) regime for one ray refracting through a glass sphere in water with T~1.

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

and

Rayleigh limit

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 (MD theory)

implies

Then the electric field Eout is proportional to as a result of the Abbe sine condition. The fields are then:

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.

Relation between Debye potentials and the fields potentials below

Total fields: internal plus external

Maxwell stress-tensor

Force on the sphere

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 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.

Mirror method

Take the objective and replace it

by a mirror .

Dual objective method 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.

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

Results 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.

Comparison between MDSA theory and

experiment. Effects of limited objective

filling factor, and spherical aberration

clearly appearing.

Multiple minima due to spherical aberration 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.

Viana, Rocha, Mesquita, Mazolli,

Maia Neto and Nussenzveig, PRE (2007).

Experimental implementation of optical tweezers 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.

Schematic set-up

Set – up at UFMG 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.

Brownian motion of a microsphere in a harmonic potential 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.

Langevin equation:

Position correlation function satisfies the equation:

Neglecting inertia and using the equipartition theorem

Back-scattering profile 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.

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

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 same diameter, , as in the previous slide.

(mm)

This effect has potential application in colloidal physics, same diameter, , as in the previous slide.

for determination of refractive index of coloidal particles and studies of coloidal growth.

Calibration same diameter, , as in the previous slide.

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.

For an expansion around x which is the case if the incident beam on the sphere has radial simmetry.i = 0 then,

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 scattering

From the time constant scattering

From <x2>

Since from the measurements one can get the stiffness k and the

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 .

Viana, Teixeira, and Mesquita, PRE (2002) the

which agrees within 5% with the expected value for this bead in water.

Summary the

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.

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