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4.2 Theoretical Analysis I – Optical Torque 133
Eﬀect of Light Beam Proﬁles

The trappingforce and rotation rate are calculated for various beam intensity
proﬁles of the form

2
2
TEM 00 (Gaussian) mode : I(r)= I 0 exp(−2r /w ), (4.16)
0
uniformly ﬁlled aperture mode : I(r)= I 0 , (4.17)
2
2
2
TEM 01 (donut) mode : I(r)= I 0 (r/w 0 ) exp(−2r /w ). (4.18)
0
The fractions of the total beam power that enters the lens aperture are 87%
(a), 100% (b), and 59% (c). Below, all the light powers are normalized after
the lens aperture.
Example 4.1. Analyze the relationship between the rotation rate and the slope
angle with illuminated by beams with Gaussian (a), uniformly ﬁlled (b), and
donut (c) intensity proﬁles.
Solution. Figure 4.15 shows the numerical results for the rotor of d =
3 µm,h =10 µm, and n 2 =1.53 in water of µ = 1 mPa s at the laser power of
P = 100 mW. Since the torque at an arbitrary point is deﬁned as the product
of force F t and distance r from the rotation axis, a high rotation rate (torque
eﬃciency) can be obtained with beam proﬁle (c) which has high intensity at
the outer part of the aperture. Optical torque increases in linear proportion
to the radius of the rotor, and the dragforce increases as twice the radius,
therefore, rotation rate decreases inversely proportional to the radius. Rota-
tion rate increases as the slope angle a increases and reaches a maximum at
a =75 . As a result, a laser beam with a strongproﬁle at the outer part and
◦
a rotor with small radius are eﬀective for increasingthe rotation rate.
Example 4.2. Analyze the relationship between torque eﬃciency Q torque and
beam waist position z f for the rotor illuminated by converging laser beams of
the intensity proﬁles with (a)–(c).

2,500
(c)
2,000 (a) Gaussian (b)
Rotation rate (rpm) 1,500 (c) Donut (a)
(b) Uniform

1,000
500

0
0 30 60 90
Slope angle (Deg)
Fig. 4.15. Relationship between rotation rate and slope angle of the rotor of d =
3 µm,h =10 µm, and n 2 =1.53 in water at P = 100 mW for the Gaussian (a),
uniformly ﬁlled (b), and donut (c)beam intensity proﬁles

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