THORLABS EDU-OT3, User Manual

Page: 9 / 94

Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers
Page 6
MTN024417-D02
Here,
𝑐𝑐
is the speed of light in vacuum and
𝐼𝐼
is the incident intensity and
𝜎𝜎
is the scattering
cross-section of the incident light. It is important to note that the scattering force is
proportional to the intensity and points in the beam direction.
3.2. Geometrical Optics Approach in the Mie Regime R >>
λ
This ray optical approach deals with the second possible extreme case. Here, we will
assume that the radius of the bead is much larger than the wavelength of the incident
laser. In this range, the conditions of geometric (ray) optics are fulfilled and one can think
of the laser beam as a bundle of rays. Typically, this assumption is valid for beads with a
radius
𝑅𝑅
that is greater than
∼ 10 𝜆𝜆
. The basics of the theoretical derivation of the gradient
and scattering force in accordance with this model can be found in a work by Ashkin 5 .
The particle properties of light must now be taken into account, namely that light can
transfer momentum to an object in the form of photons. The force action of a beam on a
particle can be explained using Newton’s second law: the force on a particle is exactly
equal to the change in the momentum of the particle over time:
𝐹𝐹 =
𝑑𝑑𝑝𝑝
𝑑𝑑𝑡𝑡
(9)
The following equation describes the change in momentum
𝑑𝑑𝑝𝑝
𝑑𝑑𝑡𝑡
of a beam over time in a
medium with a refractive index
𝑛𝑛
𝑚𝑚
:
𝑑𝑑𝑝𝑝
𝑑𝑑𝑡𝑡
=
𝑛𝑛
𝑚𝑚
𝑐𝑐
𝑃𝑃
𝐵𝐵𝑆𝑆𝐺𝐺𝑚𝑚
=
𝑛𝑛
𝑚𝑚
𝑐𝑐
𝐼𝐼
(
𝑟𝑟
)
𝑑𝑑𝑑𝑑
(10)
Here,
𝐼𝐼 ( 𝑟𝑟 )
is the intensity distribution in the beam cross-section. Often, a Gaussian profile
is used, in which the intensity decreases in a Gaussian distribution from the center of the
beam outward. This is also the case in our setup.
If a beam with the power
𝑃𝑃
𝐵𝐵𝑆𝑆𝐺𝐺𝑚𝑚
hits a sphere at an angle of
𝜗𝜗
, part of the beam will be
reflected and part of it will reach the interior of the sphere through transmission (see Figure
2). For the power of these two partial beams, the following is in effect:
𝑃𝑃
𝑟𝑟𝑆𝑆𝑟𝑟𝑟𝑟
= 𝑃𝑃
𝐵𝐵𝑆𝑆𝐺𝐺𝑚𝑚
⋅ 𝑅𝑅
𝑟𝑟
(11)
𝑃𝑃
𝑡𝑡𝑟𝑟𝐺𝐺𝑛𝑛𝑡𝑡
= 𝑃𝑃
𝐵𝐵𝑆𝑆𝐺𝐺𝑚𝑚
⋅ 𝑇𝑇
(12)
Here,
𝑅𝑅
𝑟𝑟
is the reflectivity and
𝑇𝑇
is the transmissivity. The transmitted beam transports
momentum into the sphere in accordance with the equation (10).
5
Ashkin A., Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime.
In: Biophys. J. 61 (1992) 2, 569-582