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Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers
Page 4
MTN024417-D02
Chapter 3 Principles of Optical Tweezers
To describe the function of optical tweezers, we will examine the force that a focused laser
beam with a Gaussian intensity profile (the TEM
00
mode) exerts on an object, which is near
or in the focus. Usually one also assumes that the object is a bead, which consists of a
dielectric, linear, isotropic, and spatially and chronologically non-dispersive material. In the
experiments described below, micron-sized beads made of polystyrene are primarily used.
It is customary to describe the force of the laser on the object by separating it into two
components. One component, the scattering force, acts along the direction of beam
propagation. The second component acts along the intensity gradient and is therefore
called the gradient force. The gradient force can act in different directions with respect to
the beam. As the laser has a Gaussian intensity profile, the gradient force can act
orthogonally to the beam, but it can also act parallel to the beam, as the laser is focused
and therefore also has an intensity gradient along the beam axis. These two components
and their relationship to one another are the defining factors for whether or not a particle
can be trapped by the optical trap. Stable optical tweezers are only obtained if the gradient
force, which pulls the object in the direction of the focus, is greater than the scattering
force, which pushes the particle in the direction of the beam away from the focus.
The various theoretical approaches to describe optical trapping can roughly be divided
according to the areas in which they are valid. The relationship of the radius
𝑅𝑅
(or diameter
𝑑𝑑
) of the bead to the wavelength
𝜆𝜆
of the incident laser beam is the dividing factor. The
case
𝑅𝑅 ≈ 𝜆𝜆
is theoretically very complex and shall therefore not be dealt with here. The
two extreme cases for very large and very small particles are summarized below:
3.1. Dipole Approach in the Rayleigh Scattering Regime R <<
λ
The first case we will consider is when the radius
𝑅𝑅
of the bead is significantly smaller than
the wavelength
𝜆𝜆
of the incident laser beam. Then, the electrical field
𝐸𝐸
�⃗
( 𝑟𝑟
⃗
)
is approximately
spatially constant with respect to the particle and the situation can be portrayed as follows:
As the bead is assumed to be dielectric, one can imagine it as a collection of
𝑁𝑁
point
dipoles. Due to their polarizability, a dipole moment
𝑝𝑝
⃗
𝑖𝑖
is induced in each of the point
dipoles by the incident laser beam. Due to the linearity of the material, the following applies:
𝑝𝑝
⃗
𝑖𝑖
= 𝛼𝛼 ⋅ 𝐸𝐸
�⃗
(
𝑟𝑟
⃗
𝑖𝑖
) (1)
Here,
𝑟𝑟
⃗
𝑖𝑖
is the location of the
i-th
point dipoles and
𝐸𝐸
�⃗
(
𝑟𝑟
⃗
𝑖𝑖
)
is the electrical field strength at
this location. In addition, the electrical field of the laser appears to be approximately
spatially constant for the bead due to the condition
𝑅𝑅 ≪ 𝜆𝜆
, meaning that at a certain point
in time
𝑡𝑡
0
the strength of the electrical field is equally great for all point dipoles of the bead.
As a result, the induced dipole moment is equally great for all
𝑁𝑁
point dipoles. The
polarization
𝑃𝑃
⃗
resulting from the induced dipole moments is then