Universidade Federal de Minas Gerais Belo Horizonte - MG - Brazil Department of Physics

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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 UFMG

Oscar N. Mesquita Departamento de Física, ICEX, Universidade Federal de Minas Gerais Belo Horizonte, Brasil Prof. Ubirajara AgeroProf. 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 I Summer School on Optics and Photonics January of 2010, Concepción, Chile

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 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.

Schematic set-up of optical tweezers Lecture 1 Optical tweezers: basic concepts and comparison between experiments and an absolute 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 Where 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 n H = 1.33 and n V = 1.50 are the index of refraction of the water and glass. For Pot = 1 mW and  = 45 o, the gradient force is F g = 0.95 pN. For small displacements, sin  ~  and sin  ~ , with  ~. Then and

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: 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 objective Abbe sine condition for objectives (minimum aberration):

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

Incident beam 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 J M 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 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 numerical aperture 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. T 0 is the laboratory temperature; T is the boiling temperature of water when the bead jumps; R is the radius of the Hg droplet; P a is the absorbed power; P L is the local power we want; A is the absorption coefficient of Hg. A=0.272 for =832nm P L =A.P a

Measurement of the beam profile entering the objective Mirror method 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) Dual objective method

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 potential Langevin equation: Position correlation function satisfies the equation: Neglecting inertia and using the equipartition theorem

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,, 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)

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. Calibration

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 i = 0 then, In this case the intensity correlation function is related to the second order correlation of bead center of mass position.

First ( ) and second order ( ) 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 x 0 = 0 in the backscattering profile has twice the frequency of g (2) (t) for the bead at x 0 > 0.

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

From the time constant From

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