Page 106 -

96 3 Optical Tweezers
2
2
The equation of the microsphere located on the z-axisis(z −s) +y = r 2 0
where r 0 is the radius of the microsphere and s is the distance between the
center of the microsphere and the beam waist. From the two equations given
carlier, the intersection point α between the ray and the sphere surface is

2 2 2 2 2 2 2 2 2 2 2
2sZ − 4s Z − 4Z (s − r + t ω )(Z + t ω )
0
0
0
0
2
0
2
α = 2 2 2 .
2(Z + t ω )
0 2
Accordingto the Pythagoras theorem

2
2
β = r − (s − α) .
0
The incident angle θ 1 of a Gaussian ray enteringthe sphere at the inter-
section point (α, β) is the angle between the tangential vector a of the ray
and the vector b pointingfrom the point (α, β) to the center of the sphere is
ab
θ 1 = arccos ,
|a|·|b|
where a =(1,f(t, α)),f is the derivative function of y, that is
tω 2 α
f(t, α)= ,
Z 2 1+ α 2
0 Z 2
0
b =(s − α, −β).
Here
θ 2 =arcsin{(n 1 /n 2 )sin θ 1 },

2
2
1 tan(θ 2 − θ 1 ) sin(θ 2 − θ 1 )
R(t, s)= + ,
2 tan(θ 2 + θ 1 ) sin(θ 2 + θ 1 )
and T =1 − R.
The trappingefficiencies Q s and Q g are given from (3.5) and (3.6) as
2
T {cos(2θ 1 − 2θ 2 )+ R(t, s) cos(2θ 1 )}
Q s =1 + R(t, s) cos(2θ 1 ) − 2 ,
1+ R(t, s) +2R(t, s) cos(2θ 2 )
2
T {sin(2θ 1 − 2θ 2 )+ R(t, s) sin(2θ 1 )}
Q g = R(t, s) sin(2θ 1 ) − 2 .
1+ R(t, s) +2R(t, s) cos(2θ 2 )
Consideringthe z-component,

1

Q =Q s cos φ, cos φ = ,
s
1+ f(t, s) 2
f(t, s)

Q = Q g sin τ, .
g sin τ = 2
1+ f(t, s)

The trappingefficiency alongthe z-axis due to a ray is given as Q z = Q +Q .

g
s