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3.2 Theoretical Analysis 87
As a consequence, the momentum change along the transverse direction before
and after the ray incidence is expressed as
M i sin θ 1 − (M r sin θ 1 + M t sin θ 2 )= M i sin θ 1 (1 − R − T)=0.

Example 3.2. Show that the optical pressure force exerted by a ray incidence
at the interface is given by (3.1).
Solution. The vertical components of the momentum of incidence, reflection,
and refraction rays are
M i cos θ 1 ,
M r cos θ 1 = RM i cos θ 1 ,
M t cos θ 2 =(n 2 /n 1 )TM i cos θ 2 .

The momentum change along the vertical direction before and after the ray
incidence is

M i cos θ 1 − (−M r cos θ 1 + M t cos θ 2 )= M i [(1 + R)cos θ 1 − (n 2 /n 1 )T cos θ 2 ].
Since M i = n 1 P/c, the optical pressure F is given as (3.1).
In summary, a light momentum change ∆M per second through reflection,
refraction, scatteringand absorption causes the optical pressure F. F increases
as the incident angle increases, leadingto the liftingof the microobject if laser
light is strongly focused by an objective lens with a high NA.
The principal relationship between F and the power P is
F = Q(n 1 /c)P, (3.4)

where Q is a nondimensional efficiency parameter (called trappingefficiency)
and n 1 P/c is the incident momentum per second of a ray of the power P in
a medium of the refractive index of n 1 . The trappingefficiency Q depends on
not only the object shape and refractive index but also the optical parameters
of the trappinglaser beam. It has the maximum value of 2 when all incident
rays are reflected from a totally reflectingmirror.

3.2.2 Optical Trapping Efficiency
Qualitative optical trappingdescription is possible for a small transparent
particle whose refractive index is slightly higher than that of the surround-
ingmedium. Figure 3.8a shows that the focused laser beam illuminates the
upper part in a microsphere. Consider a typical pair of rays a and b of the
focused beam under the assumption of zero surface reflection. When an inci-
dent ray refracts at the top surface, the momentum changes and the upward
optical pressure force (perpendicular to the sphere surface) F at is exerted.
The optical pressure force F ao is also exerted when ray a is emitted from the
bottom surface. The sum of the both pressure force leads to force F a owing
to ray a. Similarly, ray b produces the pressure force F b .Thesumof F a