THORLABS EDU-OT3, User Manual

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Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers
Rev B, February 15, 2022
Page 5
𝑃𝑃
⃗
=
1
𝑉𝑉
�
𝑝𝑝
⃗
𝑖𝑖
=
𝑁𝑁
𝑉𝑉
⋅ 𝛼𝛼 ⋅ 𝐸𝐸
�⃗
= 𝜒𝜒 ⋅ 𝜖𝜖
0
⋅ 𝐸𝐸
�⃗
(2)
where
𝜒𝜒
is the electrical susceptibility,
𝜖𝜖
0
is the electrical constant, and
𝑉𝑉
is the volume of
the bead. The potential energy
𝑈𝑈
𝑖𝑖
of one of the point dipoles with dipole moment
𝑝𝑝
⃗
in the
electrical field
𝐸𝐸
�⃗
is
𝑈𝑈
𝑖𝑖
= −𝑝𝑝
⃗
𝐸𝐸
�⃗
. Because there are
𝑁𝑁
point dipoles in a bead with the volume
𝑉𝑉
, the energy density in the bead is defined by:
𝑈𝑈 =
𝑁𝑁 ⋅ 𝑈𝑈
𝑖𝑖
𝑉𝑉
= −
𝑁𝑁
𝑉𝑉 ⋅ 𝑝𝑝⃗
�
𝑃𝑃�⃗
⋅ 𝐸𝐸�⃗ = −𝑃𝑃�⃗ ⋅ 𝐸𝐸�⃗
(3)
The occurrence of the gradient force, which is a force component that is directed in the
direction of the intensity gradient of the incident electrical field, can be explained when one
observes this potential energy
𝑈𝑈
of the bead in the electrical field. Equation (2) states that
𝑃𝑃
⃗
is proportional to
𝐸𝐸
�⃗
. Therefore, according to equation (3),
𝑈𝑈
is proportional to
�𝐸𝐸�⃗�
2
and
thus to intensity
𝐼𝐼 ∝ �𝐸𝐸�⃗�
2
of the incident field. The force exerted on the particle by the
incident field is proportional to the gradient of the potential energy
∇𝑈𝑈
and therefore
proportional to the intensity gradient
∇𝐼𝐼
. The following equations describe the gradient
force:
𝐹𝐹
𝐺𝐺𝑟𝑟𝐺𝐺𝑑𝑑
=
2 𝜋𝜋 𝛼𝛼
𝑐𝑐 𝑛𝑛
𝑚𝑚
2
∇𝐼𝐼
(4)
𝛼𝛼 = 𝑛𝑛
𝑚𝑚
2
𝑅𝑅
3
�
𝑚𝑚
2
− 1
𝑚𝑚
2
+ 2
�
(5)
𝑚𝑚 =
𝑛𝑛
𝑝𝑝
𝑛𝑛
𝑚𝑚
(6)
Here,
𝛼𝛼
is the polarizability of the dipoles and
𝑚𝑚
is the relationship of the refraction index
of the particles,
𝑛𝑛
𝑝𝑝
(polystyrene in our case) to the refraction index of the surrounding
medium,
𝑛𝑛
𝑚𝑚
(water in our case).
The destabilizing scattering force component is explained by the scattering of the incident
light at the particle. The force action is created by the absorption and isotropic re-emission
of the light by the bead. As
𝑅𝑅 ≪ 𝜆𝜆
, the conditions are fulfilled for Rayleigh scattering. The
resulting force can be stated as follows:
𝐹𝐹
𝑆𝑆𝑐𝑐𝐺𝐺𝑡𝑡𝑡𝑡𝑆𝑆𝑟𝑟𝑖𝑖𝑛𝑛𝑆𝑆
=
𝜎𝜎 𝑛𝑛
𝑚𝑚
𝑐𝑐
⋅ 𝐼𝐼
(7)
𝜎𝜎 =
128 𝜋𝜋
5
𝑅𝑅
6
3 𝜆𝜆 4
�
𝑚𝑚
2
− 1
𝑚𝑚 2 + 2
�
2
(8)