THORLABS EDU-OT3, User Manual

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Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers
Page 12
MTN024417-D02
The total force
𝐹𝐹
𝑡𝑡𝑚𝑚𝑡𝑡
is the sum of the total scattering force
𝐹𝐹
𝑠𝑠 , 𝑡𝑡𝑚𝑚𝑡𝑡
and the total gradient force
𝐹𝐹
𝑔𝑔 , 𝑡𝑡𝑚𝑚𝑡𝑡
, where
𝐹𝐹
𝑠𝑠 , 𝑡𝑡𝑚𝑚𝑡𝑡
= � � 𝐹𝐹
𝑠𝑠
∙ cos 𝜙𝜙𝑟𝑟𝑑𝑑𝑟𝑟𝑑𝑑𝛽𝛽
2𝜋𝜋
0
𝑚𝑚
𝑚𝑚𝑚𝑚𝑚𝑚
0
(25)
and
𝐹𝐹
𝑔𝑔 , 𝑡𝑡𝑚𝑚𝑡𝑡
= � � −𝐹𝐹
𝑔𝑔
∙ sin 𝜙𝜙𝑟𝑟𝑑𝑑𝑟𝑟𝑑𝑑𝛽𝛽
2𝜋𝜋
0
𝑚𝑚
𝑚𝑚𝑚𝑚𝑚𝑚
0
(26)
In Figure 7, we plotted the gradient force, the scattering force and the resulting total force
as a function of the focus position
𝑆𝑆
relative to the radius
𝑅𝑅
of the sphere. The direction is
the Z axis, meaning that the plot shows the behavior when the focus is moved through the
center of the sphere from bottom to top. The curves were calculated with the above
equations with a set of typical parameters since we only want to discuss the curve’s
general form. Therefore, the forces are given in arbitrary units (a.u.). On the horizontal
axis, “+1” corresponds to the focus position on the sphere’s outer surface right above the
sphere’s center while “-1” corresponds to the lower surface below the center.
Overview of All Forc
es When Focus is Changed in Z-Direction 6
From Figure 7, we learn that
1. The gradient force always points towards the focus. For example, when
𝑆𝑆 / 𝑅𝑅 ≥ 0
,
the focus is above the sphere’s center. Then
𝐹𝐹
𝑔𝑔
≤ 0
and the gradient force pushes
the sphere upwards into the focus.
2. The scattering force always points in the direction of propagation of the incident
beam, in this case downwards. Note that
𝐹𝐹
𝑠𝑠
≥ 0
in the entire range of the plot.
Fo
rc
e [a.u.]
Gradient force,
𝐹𝐹
𝑔𝑔 , 𝑡𝑡𝑚𝑚𝑡𝑡
Scattering force,
𝐹𝐹
𝑠𝑠 , 𝑡𝑡𝑚𝑚𝑡𝑡
Total force,
𝐹𝐹
𝑡𝑡𝑚𝑚𝑡𝑡
Location of the focus in units of the sphere radius
𝑅𝑅
(Z direction)