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86 3 Optical Tweezers
Incident light Reflected light

q 1 q 1
n 1
M r
n 2

q 2 M i
M t

Refracted light
Fig. 3.7. Relationship between incident, reflected, and refracted rays at interface

The optical pressure force F exerted due to the reflection and refraction
at the interface is given by (3.1) considering the momentum change in the
vertical direction (Example 3.2).

n 2 n 1 P
F = (1 + R)cos θ 1 − T cos θ 2 , (3.1)
n 1 c
where c is the speed of light in vacuum, T and R are the Fresnel transmission
and reflection coefficient, respectively. In the case of circularly polarized light,
R is given as the average of R s for s-polarization and R p for p-polarization
leadingto (3.2)

2 2
1 1 tan (θ 2 − θ 1 ) sin (θ 2 − θ 1 )
R = (R s + R p )= + . (3.2)
2
2
2 2 tan (θ 2 + θ 1 ) sin (θ 2 + θ 1 )
Since no absorption is assumed
T =1 − R. (3.3)
The total optical pressure actingon a microobject is the vector sum of the
force over the entire cross-section.
Example 3.1. Show that optical pressure is perpendicular to the surface.

Solution. When a ray with a momentum of M i is incident to the interface be-
tween index of n 1 and n 2 with an angle of θ 1 , the momentums of reflection M r
and refraction M t areexpressedas M r = RM i and M t =(n 2 /n 1 )TM i which
comes from the speed of light c/n and the light energy Mc/n. The transverse
components of the momentum of incidence, reflection, and refraction rays are

M i sin θ 1
M r sin θ 1 = RM i sin θ 1 ,
M t sin θ 2 =(n 2 /n 1 )TM i sin θ 2 = TM i sin θ 1 .