Universidade Federal Universidade Federal de Minas Geraisde Minas Gerais

Universidade Federal Universidade Federal de Minas Geraisde Minas Gerais

Belo Horizonte - MG - BrazilBelo Horizonte - MG - Brazil

Department of Physics Department of Physics

Belo HorizonteBelo HorizonteBelo HorizonteBelo Horizonte

LocalizationLocalization

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.

Welcome to UFMGWelcome to UFMGWelcome to UFMGWelcome to UFMG

Oscar N. MesquitaDepartamento 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)

ColaboratorsProf. 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)

SponsorsFapemig, CNPq, Finep, Instituto do Milênio de Nanotecnologia, Instituto do Milênio de Óptica Não-linear, Fotônica e Biofotônica e Instituto Nacional de Fluidos Complexos

I Summer School on Optics and Photonics19-22 January of 2010, Concepción, Chile

Lecture 1Optical tweezers: basic concepts and comparison between experiments and

anabsolute theory

This lecture will be largely based on the articles by A. Ashkin, by Mazolli, Maia Netoand Nussenzveig and on the doctorate thesis of Alexander Mazolli (UFRJ, 2003) and doctorate

thesis of Márcio Santos Rocha (UFMG, 2008).

Lecture 2Application 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 3Defocusing Microscopy: a new way of phase retrieval and 3D imaging of

transparent objectsDefocusing 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 4Application 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.

Schematic set-up of optical tweezers

Lecture 1Optical tweezers: basic concepts and comparison between

experiments and anabsolute theory

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

Geometric optics description (<<a)

Figures below are from Mazolli’s thesis

Cilindrical Beam – refracted (gradient force)

Centered (refracted)

Momentum conservation

Light carries momentum

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)

(focus below the sphere center)

Out-of-center (refracted)

Centered (refracted)

(focus above the sphere center)

Conical Beam – reflected (radiation pressure)

Rayleigh limit description (>>a)Electric dipole in an inhomogeneous electric field

Only the gradient force exists in this limit

Consequently the stiffness K is proportional to 3a

aWhere is the particle radius

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 = 45o, the gradient force is Fg = 0.95 pN.For small displacements, sin ~ and sin~, with ~ .Then

)/( ax

a

x

n

n

c

PotnF

v

HHR 12 and

a

K1

Important scalings

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:

)/( axFFx

For small displacements in relation to the equilibrium position

a

xFx and

a

K1

Rayleigh limit

3aK

a

a

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 condition2) Richards-Wolf approach

Gaussian laser incident field

objective

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:

with implies

Incident beambefore 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.

jMMd ,

Total fields: internal plus external

Maxwell stress-tensor

Force on the sphere

Relation between Debye potentials and the fields

Results

Microbolometer

Viana, Mesquita & Mazolli, APL 81, 1765 (2002)

Measurement of the local power at the focus of a high numericalaperture objective

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.

T0 is the laboratory temperature;T is the boiling temperature of waterwhen 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.

A=0.272 for =832nm

PL=A.Pa

Measurement of the beam profile entering the objective

Mirror methodTake the objective and replace it

by a mirror .

Standard method to measurethe local power at the focusof high numerical apertureobjetives. One has to becareful because the transmissioncoefficients of objectives in theIR are not spatially uniform,and changes the beam profile, as shown by Viana, Rocha,Mesquita, Mazolli, and MaiaNeto, Appl. Opt. 45, 4263 (2006)

Dual objective method

-1

0

1

2

3

4

0 0.5 1 1.5 2 2.5 3

/

PL (p

N/

m m

W)

a (m)

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 clearly appearing.

Results

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

potentialLangevin equation:

0)()0()()0()()0(

2

2

txxkdt

txxd

dt

txxdm

)(2)()()(2

2

ttTktftftfkxdt

dx

dt

xdm B

Position correlation function satisfies the equation:

Neglecting inertia and using the equipartition theorem

i

B

k

Tkx 2

iii

t

i

Bii

k

a

k

ek

Tktxx i

6

0

Back-scattering profileOne 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 backscatteringprofile with the detector in thecenter.Losanges are the backscatteringprofile with the detector movedto maximize the intensity ofone of the lateral peaks.

As compared to the previousslide, the central peak nowhas minimum intensity. This effect and the lateral peaks can be explained by theMD theory.

%57.2 m

Mazolli, Maia Neto and Nussenzveig - MD theory

(m)

0

20

40

60

80

100

0 20 40 60 80 100 120 140 160

Backscattering intensity of droplets of CCl4

in water.

The oscilations are due to interference as the size of the droplets changes in time.

I (k

Hz)

time (s)

This effect has potential application in colloidal physics,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.

xfeIxI 0

Calibration

)()()( 000

0

0

ii

xiiii

xfi

xxdx

dIxIxxI

eIxI

i

i

)(

001)(

)()0(

0

2222

0

)2(

iii

zxi

xxx

tzztxxxI

tIItg

i

i x

f

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.

2

)()()(

20

2

2

00

0

ii

xi

iii

xx

dx

IdxIxxI

i

)(

001)(

)()0(

0

22222222

0

)2(

iii

zxi

xxx

tzztxxxI

tIItg

For an expansion around xi = 0 then,

In this case the intensity correlation function is related to the second order correlation of bead center of mass position.

000

ixidx

dI

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.

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.

Correlation function obtained in the linear part of the scatteringprofile, where clearly two time constants appear: a shorter one for motion perpendicular and the longer one parallel to the incident direction.

Results

cmdynkx /0002.00058.0

cmdynkx /0003.00059.0

From the time constant

From <x2>

Parallel Stokes friction near a wall (Faxen’s expression)

1543

0

//

16

1

256

45

8

1

16

91

h

a

h

a

h

a

h

a

Since from the measurements one can get the stiffness k and thefriction coefficient , one can check how the friction changes asthe bead approaches the glass slide. One can move the bead in relation to the glass slide by just moving the objective.

where is the radius of the bead, h is the distance from its center-of-mass to the glass slide, and .a 60

a

sg /1027.2 50

Viana, Teixeira, and Mesquita, PRE (2002)

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 3m 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 accuratelymeasure the stiffness of the the optical tweezers.