THORLABS EDU-OT3, User Manual

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Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers
Page 14
MTN024417-D02
Influence of the Numerical Aperture
As we have seen above, the angle of incidence of the partial rays plays a crucial role in
optical tweezing. The angle is defined by the numerical aperture of the objective: the
numerical aperture ( ) describes the acceptance cone of an objective and is given by
= 𝑛𝑛 ∙ sin 𝜙𝜙
(27)
where
𝑛𝑛
is the refractive index of the material between the objective and the focus and
𝜙𝜙
is half of the angle of the maximum light cone.
As discussed, the gradient force
𝑔𝑔 ,
needs to exceed the scattering force
𝑠𝑠 ,
to get a stable trap. Next, we want to investigate a measure for the trap’s
strength. For that, we can have a look at the ratio of
𝑔𝑔 ,
and
𝑠𝑠 ,
at the
point
𝑆𝑆 / 𝑅𝑅 = 1
since we have shown that scattering and gradient force are strongest
when the laser focus is at/near the sphere’s surface (i.e.,
|
𝑆𝑆 ⁄𝑅𝑅
|
= 1
). So for a stable
trap, we can assume the condition 7 :
�
𝑔𝑔 ,
( 𝑆𝑆 ⁄
𝑅𝑅 = 1)
𝑠𝑠 ,
( 𝑆𝑆 ⁄
𝑅𝑅 = 1)
� ≥ 1
(28)
Absolute values are used since
𝑔𝑔 ,
is negative. Figure 9 shows how this strength
depends on the numerical aperture. Again, we plotted a curve with typical parameters to
show the general behavior of the curve; therefore, we do not focus on the concrete
numbers.
Figure 9 shows the fundamental behavior that the strength of the trap increases with
increasing numerical aperture. Also, it becomes apparent that there is a lower limit for the
numerical aperture of the objective. Below that, no trapping occurs since the gradient force
never exceeds the scattering force.
Behavior of the Trap’s Strength with Respect to the Numerical Aperture
6
7
The absolute value of the forces is used since
𝑔𝑔 ,
is negative.
|
𝑔𝑔
,
/
𝑠𝑠
,
|
Numerical aperture
NA = n . sin(phi)
F
g,tot
F
g,tot
F
g,tot
F
s,tot
F
s,tot
F
g,tot
( S / R =1)
F
s,tot
( S / R =1)
1
| F
g,tot
/ F
s,tot
|