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Portable Optical Tweezers Kit Chapter 4: Principles of Optical Tweezers
Rev B, July 8, 2019 Page 9
𝑄
𝑔
= 𝑅
𝑟
sin
(
2𝜃
)
−
𝑇
2
(
sin
(
2𝜃 − 2𝑡
)
+ 𝑅
𝑟
cos
(
2𝜃
))
1 + 𝑅
𝑟
2
+ 2𝑅
𝑟
cos
(
2𝑡
)
(16)
Here,
𝑡
is the angle at which the first transmitted beam is refracted toward the normal (see
Figure 2
). According to Snell’s law of refraction, the following relationship is in effect for
the angles
𝜃
and
𝑡
:
sin
(
𝜃
)
sin
(
𝑡
)
=
𝑛
𝑝
𝑛
𝑚
(17)
𝑄
𝑠
and 𝑄
𝑔
are dimensionless Q factors, which state what percentage of the incident
momentum contributes to the force parallel or perpendicular to the beam, respectively.
These factors depend heavily on the angle of incidence of the beam, as one can see from
the equations. This angle becomes larger the more heavily the beam is focused, which
occurs when a higher numerical aperture objective is used.
The component of the beam that points in the incident direction (Z direction) ultimately
causes the scattering force
𝐹
𝑠
. The component perpendicular to this (Y direction) is mainly
responsible for the gradient force
𝐹
𝑔
.
In order to obtain the overall power, one must
naturally consider all partial beams and integrate all of them. That will be discussed in
detail below.
Q Factor Angular Dependence
5
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