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3.2 Theoretical Analysis 89
PR
Incident light q1
P a
q2 a+b
2
PR T 2
F s
o PT 2
F g b
x
b
PRT 2 y

Fig. 3.9. Optical pressure force on microsphere exerted by single incident ray [3.4]

Example 3.3. Show that the optical pressure forces exerted by a ray of the
power P incident to a dielectric microsphere of the refractive index n 2 at the
angle θ 1 surroundingthe medium of n 1 are given by (3.5) and (3.6).

Solution. When the ray strikes the sphere, a fraction of the light PR is re-
flected and the remainder is transmitted into the sphere, as shown in Fig. 3.9.
No absorption is assumed for the sphere. The transmitted fraction produces
an infinite number of internal reflections within the sphere and infinite num-
bers of scattered rays escape the sphere. A series of scattered ray powers are
expressed as

2
2
2
n
PR, PT ,PRT ,...,PR T ,... (n =0, 1, 2, 3,...)
where T and R are the Fresnel transmission and reflection coefficients on the
surface at θ 1 , respectively. The angles of the scattered rays with the incident
ray are expressed as
π +2θ 1 ,α,α + β, α +2β, ··· ,α + nβ, ··· (n =0, 1, 2, 3, ),

where α =2(θ 1 − θ 2 )and β = π − 2θ 2 .
Since the optical pressure force in the x-direction is defined as the mo-
mentum change per second due to the scattered rays

∞
n 1 P n 1 PR n 1 P n 2
Fs = − cos(π +2θ 1 )+ R T cos(α + nβ) ,
c c c
n=0
where n 1 P/c is the incident light momentum per second in the x-direction.
Similarly, for the y-direction,

∞
n 1 PR n 1 P n 2
Fg =0 − sin(π +2θ 1 )+ R T sin(α + nβ) .
c c
n=0
Here introducingthe followingequations to the sum over n in the above
equations lead to (3.5) and (3.6).