# Optical Tweezers

Entry by Haifei Zhang, AP 225, Fall 2009

## Contents

## What is optical tweezers

Optical Tweezers use light to manipulate microscopic objects as small as a single atom. The radiation pressure from a focused laser beam is able to trap small particles. In the biological sciences, these instruments have been used to apply forces in the pN-range and to measure displacements in the nm range of objects ranging in size from 10 nm to over 100 mm.

## The physics of optical tweezers

### The basics

The most basic form of an optical trap is diagramed in Fig 1a. A laser beam is focused by a high-quality microscope objective to a spot in the specimen plane. This spot creates an "optical trap" which is able to hold a small particle at its center. The forces felt by this particle consist of the light scattering and gradient forces due to the interaction of the particle with the light (Fig 1b, see Details). Most frequently, optical tweezers are built by modifying a standard optical microscope. These instruments have evolved from simple tools to manipulate micron-sized objects to sophisticated devices under computer-control that can measure displacements and forces with high precision and accuracy.

Fig 1b shows a more detailed look at how an optical trap works. The basic principle behind optical tweezers is the momentum transfer associated with bending light. Light carries momentum that is proportional to its energy and in the direction of propagation. Any change in the direction of light, by reflection or refraction, will result in a change of the momentum of the light. If an object bends the light, changing its momentum, conservation of momentum requires that the object must undergo an equal and opposite momentum change. This gives rise to a force acting on the object.

In a typical optical tweezers setup the incoming light comes from a laser which has a "Gaussian intensity profile". Basically, the light at the center of the beam is brighter than the light at the edges. When this light interacts with a bead, the light rays are bent according the laws of reflection and refraction (two example rays are shown in Fig 1b). The sum of the forces from all such rays can be split into two components: Fscattering, the scattering force, pointing in the direction of the incident light (z, see axes in Fig 1b), and Fgradient, the gradient force, arising from the gradient of the Gaussian intensity profile and pointing in x-y plane towards the center of the beam (dotted line). The gradient force is a restoring force that pulls the bead into the center. If the contribution to Fscattering of the refracted rays is larger than that of the reflected rays then a restoring force is also created along the z-axis, and a stable trap will exist. Incidentally, the image of the bead can be projected onto a quadrant photodiode to measure nm-scale displacements .

### Ray optics explanation

In cases where the diameter of a trapped particle is significantly greater than the wavelength of light, the trapping phenomenon can be explained using ray optics. As shown in the figure, individual rays of light emitted from the laser will be refraction|refracted as it enters and exits the dielectric bead. As a result, the ray will exit in a direction different from which it originated. Since light has a momentum associated with it, this change in direction indicates that its momentum has changed. Due to Newton's laws of motion|Newton's third law, there should be an equal and opposite momentum change on the particle.

Most optical traps operate with a Gaussian beam (TEM_{00} mode) profile intensity. In this case, if the particle is displaced from the center of the beam, as in (a) in the figure, the particle has a net force returning it to the center of the trap because more intense beams impart a larger momentum change towards the center of the trap than less intense beams, which impart a smaller momentum change away from the trap center. The net momentum change, or force, returns the particle to the trap center.

If the particle is located at the center of the beam, then individual rays of light are refracting through the particle symmetrically, resulting in no net lateral force. The net force in this case is along the axial direction of the trap, which cancels out the scattering force of the laser light. The cancellation of this axial gradient force with the scattering force is what causes the bead to be stably trapped slightly downstream of the beam waist.

It must be noted that the standard tweezers works with the trapping laser propagated in the direction of gravity and the inverted tweezers works against gravity.

## The mathematics of optical tweezer

In cases where the diameter of a trapped particle is significantly smaller than the wavelength of light, the conditions for Rayleigh scattering are satisfied and the particle can be treated as a point dipole in an inhomogenous electromagnetic field. The force applied on a single charge in an electromagnetic field is known as the Lorentz force,

- <math> \mathbf{F_1}=q\left(\mathbf{E_1}+\frac{d\mathbf{x_1}}{dt}\times\mathbf{B}\right). </math>

The force on the dipole can be calculated by substituting two terms for the electric field in the equation above, one for each charge. The polarization of a dipole is <math> \mathbf{p}=q\mathbf{d}, </math> where <math> \mathbf{d} </math> is the distance between the two charges. For a point dipole, the distance is infinitesimal, <math> \mathbf{x}_1-\mathbf{x}_2. </math> Taking into account that the two charges have opposite signs, the force takes the form

- <math>

\begin{align}

\mathbf{F} & = q\left(\mathbf{E_1}\left(x,y,z\right)-\mathbf{E_2}\left(x,y,z\right)+\frac{d(\mathbf{x}_1-\mathbf{x}_2)}{dt}\times\mathbf{B}\right) \\ & = q\left(\mathbf{E_1}\left(x,y,z\right)+\left((\mathbf{x}_1-\mathbf{x}_2)\cdot\nabla\right)\mathbf{E}-\mathbf{E_1}\left(x,y,z\right)+\frac{d(\mathbf{x}_1-\mathbf{x}_2)}{dt}\times\mathbf{B}\right). \\

\end{align} </math>

Notice that the <math> \mathbf{E_1} </math> cancel out. Multiplying through by the charge, <math> q </math>, converts position, <math> \mathbf{x} </math>, into polarization, <math> \mathbf{p} </math>,

- <math>

\begin{align}

\mathbf{F} & = \left(\mathbf{p}\cdot\nabla\right)\mathbf{E}+\frac{d\mathbf{p}}{dt}\times\mathbf{B} \\ & = \alpha\left[\left(\mathbf{E}\cdot\nabla\right)\mathbf{E}+\frac{d\mathbf{E}}{dt}\times\mathbf{B}\right], \\

\end{align} </math>

where in the second equality, it has been assumed that the dielectric particle is linear (i.e. <math> \mathbf{p}=\alpha\mathbf{E} </math>).

In the final steps, two equalities will be used: (1) Vector calculus identities|A Vector Analysis Equality, (2) Faraday's law of induction|One of Maxwell's Equations.

- <math> \left(\mathbf{E}\cdot\nabla\right)\mathbf{E}=\nabla\left(\frac{1}{2}E^2\right)-\mathbf{E}\times\left(\nabla\times\mathbf{E}\right) </math>
- <math> \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t} </math>

First, the vector equality will be inserted for the first term in the force equation above. Maxwell's equation will be substituted in for the second term in the vector equality. Then the two terms which contain time derivatives can be combined into a single term.<ref> Gordon JP, "Radiation Forces and Momenta in Dielectric Media", Physical Review A (1973). 8(1): 14–21. </ref>

- <math>

\begin{align}

\mathbf{F} & = \alpha\left[\frac{1}{2}\nabla E^2-\mathbf{E}\times\left(\nabla\times\mathbf{E}\right)+\frac{d\mathbf{E}}{dt}\times\mathbf{B}\right] \\ & = \alpha\left[\frac{1}{2}\nabla E^2-\mathbf{E}\times\left(-\frac{d\mathbf{B}}{dt}\right)+\frac{d\mathbf{E}}{dt}\times\mathbf{B}\right] \\ & = \alpha\left[\frac{1}{2}\nabla E^2+\frac{d}{dt}\left(\mathbf{E}\times\mathbf{B}\right)\right]. \\

\end{align} </math>

The second term in the last equality is the time derivative of a quantity that is related through a multiplicative constant to the Poynting vector, which describes the power per unit area passing through a surface. Since the power of the laser is constant when sampling over frequencies much shorter than the frequency of the laser's light ~10^{14} Hz, the derivative of this term averages to zero and the force can be written as

- <math> \mathbf{F}=\frac{1}{2}\alpha\nabla E^2. </math>

The square of the magnitude of the electric field is equal to the intensity of the beam as a function of position. Therefore, the result indicates that the force on the dielectric particle, when treated as a point dipole, is proportional to the gradient along the intensity of the beam. In other words, the gradient force described here tends to attract the particle to the region of highest intensity. In reality, the scattering force of the light works against the gradient force in the axial direction of the trap, resulting in an equilibrium position that is displaced slightly downstream of the intensity maximum. The scattering force depends linearly on the intensity of the beam, the cross section of the particle and the index of refraction of the trapping medium (e.g. water).

## Applications

Optical Tweezers have been used to trap dielectric spheres, viruses, bacteria, living cells, organelles, small metal particles, and even strands of DNA. Applications include confinement and organization (e.g. for cell sorting), tracking of movement (e.g. of bacteria), application and measurement of small forces, and altering of larger structures (such as cell membranes). Two of the main uses for optical traps have been the study of molecular motors and the physical properties of DNA. In both areas, a biological specimen is biochemically attached to a micron-sized glass or polystyrene bead that is then trapped. By attaching a single molecular motor (such as kinesin, myosin, RNA polymerase etc.) to such a bead, researchers have been able to probe motor properties such as: Does the motor take individual steps? What is the step size? How much force can the motor produce? Similarly, by attaching the beads to the ends of single pieces of DNA, experiments have measured the elasticity of the DNA, as well as the forces under which the DNA breaks or undergoes a phase transition.

## References

[1] http://www.stanford.edu/group/blocklab/Optical%20Tweezers%20Introduction.htm

[2] http://en.wikipedia.org/wiki/Optical_tweezers#The_physics_of_optical_tweezers