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92 3 Optical Tweezers
Table 3.4. Maximum trapping efficiency for axial trap with various laser beam
profiles
beam profile downward directed upward directed
Gaussian (TEM 00)0.21 0.33
uniform 0.25 0.39
∗
donut (TEM 01 )0.26 0.41

the upward directed beam is more effective in trappingthe microsphere than
the downward-directed beam. Table 3.3 shows microsphere materials for the
analysis in this book.
The trappingefficiency dependence on the incident angle of a ray means
that trappingefficiency is related to the profile of the laser beam. Table 3.4
shows the maximum trappingefficiency calculated for input beams with
various mode intensity profiles: Gaussian, uniformly filled, and donut. The
maximum Q increases as the outer part intensity increases. Good trappingis
possible when the outer part of the aperture is filled by a high intensity to
give a laser beam with a high convergence angle.
Example 3.4. Calculate the axial trappingefficiency for a microsphere when
the focus of the uniformly input laser beam is alongthe optical axis in the
center line of the sphere.
Solution. First, we find the incident angle θ 1 (r, β)ofarayenteringthein-
put aperture of the objective lens at the arbitrary point (r, β), as shown in
Fig. 3.11a [3.4]. Since axial trapping efficiency is independent on β due to axial
symmetry, we consider r-dependence for the θ 1 (r, β). The angle φ(r) between
the incidence ray and z-axis is r 0 sin θ 1 (r)= s sin φ(r) where r 0 is the radius
of the microsphere (we take r 0 = 1 since the results in the ray optics model
are independent on r), s is the distance between the center of the microsphere
and the laser focus. From Fig. 3.11b,
r

−1
φ(r) = tan tan Φ m ,
R m
where R m is the lens radius and Φ m is the maximum convergence angle. Then
the incident angle θ 1 (r) becomes
 

2
θ 1 (r)=sin −1  sr tan Φ m 1+ r tan Φ m  .
R m R m
Next, the trappingefficiencies Q s (r)and Q g (r) are computed by the vector
sum of the contributions of all rays within the convergence angle using (3.5)
and (3.6). Here, the y-component is cancelled out due to the symmetry, only
the z-component is calculated as
Q sz (r)= Q s (r)cos φ(r),
Q gz (r)= Q g (r) sin φ(r).