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 11
Next, the forces need to be summed. For that, we start by observing Figure 6. A partial
beam with distance
𝑟𝑟
to the symmetry axis of the whole beam falls on the sphere under
an angle
𝜃𝜃
. As discussed above, we can split the resulting force in two perpendicular
components,
𝐹𝐹
𝑠𝑠
and
𝐹𝐹
𝑔𝑔
. For clarity, we can now add another partial beam, namely the one
mirrored on the symmetry axis, denoted “mirror beam”. As sketched in Figure 6, this partial
beam falls on the sphere on the other side and results in “mirrored” force vectors
𝐹𝐹
𝑠𝑠 , 𝑚𝑚𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
and
𝐹𝐹
𝑔𝑔 , 𝑚𝑚𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚
in the right part of the sphere (which were not drawn to avoid an overcrowded
figure).
The Contribution of One Ray to the Total Force 5,6
When you consider all of the partial beams with the same condition around the symmetry
axis, it immediately becomes clear that all force components in X and Y direction vanish
and only resulting force components along the Z axis remain.
These can be written as
𝐹𝐹
𝑔𝑔 , 𝑧𝑧
= −𝐹𝐹
𝑔𝑔
∙ sin 𝜙𝜙 = −𝐹𝐹
𝑔𝑔
∙
𝑟𝑟
√𝑟𝑟 2 + 𝑟𝑟 2
(22)
𝐹𝐹
𝑠𝑠 , 𝑧𝑧
= 𝐹𝐹
𝑠𝑠
∙ cos 𝜙𝜙 = 𝐹𝐹
𝑠𝑠
∙
𝑟𝑟
√𝑟𝑟
2
+ 𝑟𝑟
2
(23)
where
𝐹𝐹
𝑔𝑔 , 𝑧𝑧
is pointing in the negative Z direction and is, therefore, negative. Hence, each
infinitesimal force contributing to the total force is given by
𝑑𝑑𝐹𝐹 = 𝐹𝐹
𝑠𝑠
∙ cos 𝜙𝜙 − 𝐹𝐹
𝑔𝑔
∙ sin 𝜙𝜙
(24)
6
Adapted from A. Langendörfer: ”Aufbau einer Optischen Pinzette für das Landesmuseum für
Technik und Arbeit in Mannheim“, wissenshaftliche Arbeit, KIT, Karlsruhe, 2009