The trappingefficiency dependence on the incident angle of a ray means that trappingefficiency is related to the profile of the laser beam.. Calculate the axial trappingefficiency for a microsp
Trang 13.2 Theoretical Analysis 91 (b)
w
Fg
Fs
-1
1
O
Z
f r
f (r)
f (r)
n q1(r)
s
(a)
r
f (r)
Fm
b b
Z
Y f
O w
w '
X Y Z
Rm
Rm
Fig 3.11 Geometry for calculating axial trapping efficiency of polystyrene
mi-croshere The laser focus is on the optical axis which is parallel to the center line of the microsphere [3.4]
(a)
X Y Z
r
g a
b
Z
1 -1 f
W
W '
q1
q1
Fm
Rm
Fs
Fg
Rm
RmcotFm
n
A
(b)
W
O
f '
n
A
s ' R
Fig 3.12 Geometry for calculating the transverse trapping efficiency of polystyrene
microshere The laser focus is located along the transverse center line of the sphere [3.4]
Z
0.2 0.4 0
Trapping efficiency -0.4 -0.2
0 0.2 0.4 0.6 0.8 1.0 0.1
0.2 0.3 0.4
0
Y
Focus point
Fig 3.13 Total trapping efficiency Qt exerted on a polystyrene microsphere
sus-pended in water by trap with a uniformly filled input aperture of NA = 1.25 for
axial (a), and for transversal (b)directions
Trang 292 3 Optical Tweezers
Table 3.4 Maximum trapping efficiency for axial trap with various laser beam
profiles
the upward directed beam is more effective in trappingthe microsphere than the downward-directed beam Table 3.3 shows microsphere materials for the analysis in this book
The trappingefficiency dependence on the incident angle of a ray means that trappingefficiency is related to the profile of the laser beam Table 3.4 shows the maximum trappingefficiency calculated for input beams with various mode intensity profiles: Gaussian, uniformly filled, and donut The
maximum Q increases as the outer part intensity increases Good trappingis
possible when the outer part of the aperture is filled by a high intensity to give a laser beam with a high convergence angle
Example 3.4 Calculate the axial trappingefficiency for a microsphere when
the focus of the uniformly input laser beam is alongthe optical axis in the center line of the sphere
Solution First, we find the incident angle θ1(r, β) of a ray entering the in-put aperture of the objective lens at the arbitrary point (r, β), as shown in Fig 3.11a [3.4] Since axial trapping efficiency is independent on β due to axial symmetry, we consider r-dependence for the θ1(r, β) The angle φ(r) between the incidence ray and z-axis is r0sin θ1(r) = s sin φ(r) where r0 is the radius
of the microsphere (we take r0= 1 since the results in the ray optics model
are independent on r), s is the distance between the center of the microsphere
and the laser focus From Fig 3.11b,
φ(r) = tan −1
r
R m
tan Φm
, where Rmis the lens radius and Φmis the maximum convergence angle Then
the incident angle θ1(r) becomes
θ1(r) = sin −1
sr tan Φm
Rm
1 +
r tan Φm
Rm
2
Next, the trappingefficiencies Qs(r) and Qg(r) are computed by the vector
sum of the contributions of all rays within the convergence angle using (3.5)
and (3.6) Here, the y-component is cancelled out due to the symmetry, only the z-component is calculated as
Q sz (r) = Qs(r) cos φ(r),
Q gz (r) = Qg(r) sin φ(r).
Trang 33.2 Theoretical Analysis 93
Finally, Qsand Qg are obtained by integrating all the rays using
πR2 m
2 π
0
Rm
0
rQ sz (r)drdβ = 2
R2 m
Rm
0
rQ sz (r)dr,
πR2 m
2 π
0
R m
0
rQ gz (r)drdβ = 2
R2 m
R m
0
rQ gz (r)dr.
The total trappingefficiency is given by Q t=
Q2
s+ Q2
3.2.3 Effect of Beam Waist
In the ray optics, a laser beam is decomposed into individual rays with appro-priate intensity, direction and polarization, which propagate in straight lines
In actual conditions, the focused light beam has a beam waist, which means that each ray varies its direction near the focus Therefore, the incident angle
θ1 varies from that of the straight line, leading to the recalculation of the exact optical pressure force
We introduce a Gaussian beam profile (3.9) of a beam waist ω0 and the
depth of focus Z0 instead of straight line ray optics as
2N A , Z0= kω
2
where k is the wave number 2π/λ, λ is the wavelength, and NA is the
numer-ical aperture of the objective
To determine the incident angle θ1(r) of a Gaussian ray passingat r = r in the aperture of the objective enters at the point (α, β) on the sphere surface
as shown in Fig 3.14 The coordinates (α, β) are expressed
α =
2sZ2−
4s2Z2− 4Z2
s2− r2+
r
R m
2
ω2
Z2+
r
R m
2
ω2
2
Z2+
r
R m
2
ω2
(3.10)
β =
Then the incident angle θ1(r) is calculated as the angle between the
tan-gent vector a of the Gaussian ray at (α, β) and the direction vector b pointing
to the center of the sphere After the incident angle θ1(r) is defined, the
trap-pingefficiency alongthe optical axis can be computed Figure 3.15 show the result for a polystyrene sphere suspended in water Consideringthe beam
Trang 494 3 Optical Tweezers
y
o
Z (ab)
Fig 3.14 Geometry for calculating exact axial trapping efficiency for microsphere
considering beam waist
s
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0 0.2 0.4 0.6 0.8 1 Normalized distance between particle center and focus point
Ray optics
Diameter
20 mm 10 5 1 Beam waist
Fig 3.15 Axial trapping efficiency of polystyrene microsphere suspended in water
by converging ray approximations of straight line (ray optics)and parabolic line
(beam waist)with beam waist ω0
waist, it is seen from the figure that the axial trapping efficiency decreases to 50% that of the straight lines This is caused by the fact that focused rays are almost parallel to the optical axis near the focus, as shown in the upper left sketch in the figure
Figure 3.16 shows the transverse trappingefficiency alongthe axis perpen-dicular to the optical axis It is seen from the figure that both straight and parabolic Gaussian beam rays have almost the same numerical results This
is based on the fact that the incident angles at the surface of the sphere are almost the same for both approximations because the laser focus is located near the surface edge, maximum trapping efficiency, on the center line of the sphere (see the upper left sketch in the figure)
Example 3.5 Compute the trappingefficiency of a microsphere suspended in
water alongthe propagation axis by the laser beam emitted from the tapered
Trang 53.2 Theoretical Analysis 95
Normarized distance between microsphere center and focus point
Diameter
40 mm 10 2
20 4 Ray optics
Beam waist 0.40
0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 0.2 0.4 0.6 0.8 1.0
s
Fig 3.16 Transverse trapping efficiency of polystyrene microsphere by two
con-verging ray approximations
w1
1
R
r (z )
Fig 3.17 Geometry for calculating trapping efficiency for microsphere along
prop-agation axis by laser beam emitted from tapered lensed optical fiber
lensed optical fiber of curvature R = 10 µm, beam waist radius ω1= 5.0µm,
core refractive index n1 = 1.462, as shown in Fig 3.17 The focus distance from the tapered lensed fiber end d2and the beam radius r(z) with the beam waist ω2 are given as
d2=− n2R(n2− n1)
(n2− n1)2+ R2
λ
πω2
2, r(z) = ω2
1 +
z
kω2
2
.
Solution An equation of a ray going along the z-direction is expressed by
the variable parameter t(0 ≤ t ≤ 1) as
y = tω2
1 +
z
Z0
2
, Z0= kω22, where t = r/R mand
πω2
λ
2
n2−n1
n1R
2
+ 1
.
Trang 696 3 Optical Tweezers
The equation of the microsphere located on the z-axis is (z − s)2+ y2= r2
where r0 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
2−4s2Z2− 4Z2(s2− r2+ t2ω2)(Z2+ t2ω2)
Accordingto the Pythagoras theorem
β =
r2− (s − α)2 The incident angle θ1 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
θ1= arccos ab
|a| · |b| , where a = (1, f (t, α)), f is the derivative function of y, that is
f (t, α) = tω2α
Z2
1 +Z α22
,
b = (s − α, −β).
Here
θ2 = arcsin{(n1/n2) sin θ1}, R(t, s) = 1
2
tan(θ2− θ1)
tan(θ2+ θ1)
2
+
sin(θ2− θ1)
sin(θ2+ θ1)
2
,
and T = 1 − R.
The trappingefficiencies Qsand Qg are given from (3.5) and (3.6) as
Qs= 1 + R(t, s) cos(2θ1)− T2{cos(2θ1− 2θ2) + R(t, s) cos(2θ1)}
1 + R(t, s)2+ 2R(t, s) cos(2θ2) ,
Qg= R(t, s) sin(2θ1)− T2{sin(2θ1− 2θ2) + R(t, s) sin(2θ1)}
1 + R(t, s)2+ 2R(t, s) cos(2θ2) .
Consideringthe z-component,
Q s=Qscos φ, cos φ = 1
1 + f (t, s)2,
Q g= Qgsin τ, sin τ = f (t, s)
1 + f (t, s)2 The trappingefficiency alongthe z-axis due to a ray is given as Qz= Q + Q
Trang 73.2 Theoretical Analysis 97
Next, the trappingefficiency due to a circular element of radius β is
given as
Qc= 2πβQz
Finally, this trappingefficiency is integrated over the entire cross-section
of the sphere for all individual rays usingthe Shimpson formula under the conditions in Table 3.5
Figure 3.18 shows the axial trapping efficiency dependence on the distance
from the optical fiber end for a polystyrene sphere of radii 2.0 and 2.5µm
The laser beam profile is Gaussian and the wavelength is 1.3µm It is seen from the figure that trapping force increases as axial distance increases from zero to a beam waist of 40µm, i.e., it increases over the region in which the fiber lens is focusing, and then begins to decrease monotonically as the beam diverges beyond the focus Therefore, we can expect that the optimum dual fiber lens spacingwill exists at a point where axial trappingefficiency is changing rapidly (see Sect 3.3.4)
3.2.4 Off-axial Trapping by Solitary Optical Fiber
In recent years, studies of optical tweezers have been conducted on optical-fiber tweezers [3.12] to improve their operation in the fields of life science and
Table 3.5 Conditions for analysis of tapered lensed optical fiber trapping efficiency
refractive index
beam waist in the core (µm)5.0
radius of curvature (µm)10
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
0 50 100 150 200 250 300 350 400 Distance from fiber end (mm)
Diameter 5mm 4mm
Fig 3.18 Axial trapping efficiency dependence on distance from optical fiber end
of polystyrene sphere
Trang 898 3 Optical Tweezers
micromachines The optical fiber implementation of such tweezers is simple and inexpensive The apparatus that uses a laser diode and an optical fiber
is particularly simple since no external optics such as a dichromatic mirror, a beam splitter, and filters are required
Trappingforces can be resolved into two components: the gradient force
Fg, which pulls microspheres in the direction of the stronglight intensity,
and te scatteringforce Fs, which pushes microspheres in the direction of light propagation If a microsphere is located on the light propagation axis, the gradient forces cancel out, thereby resultingin pushingthe sphere Therefore, two counterpropagating coaxially aligned optical fibers are used to trap the sphere suspended in water [3.13] Although the sphere is stabilized axially at
a location where the scatteringforces of the two beams balance each other, the trappingin the transverse direction is weak The freedom of operation for the counterpropagating coaxially aligned optical fibers is poor In this section,
we theoretically analyze an off-axial microsphere trappingforce [3.14] in three dimensions in order to trap it with a solitary optical fiber
Analysis of Off-axial Trapping
Trappingefficiency for a microsphere on an optical axis can be calculated, from axial symmetry, as shown in Fig 3.19a, by integrating the optical pres-sure force due to an individual ray in two dimensions On the other hand, calculation in three dimensions is necessary for the off-axial trappingeffi-ciency because of axial dissymmetry Figure 3.19b shows that a ray enters at
(a)
Y
Z
Fs
Fg
Fg Fs
Total trapping force
(b) Intersection(x,y,z) Incident
angle q1
Y
Z
Beam profile
Sphere center (0,B,A)
Axial distance A Off-axial distance B
Fs
Fg
Fig 3.19 Geometry for calculating trapping efficiency for a microshere when focus
is located on optical axis (a), and at off-axis (b)
Trang 93.2 Theoretical Analysis 99
the incident angle θ1on the arbitrary intersection (x, y, z) of the surface of a sphere, whose center is located at (0, B, A) The y-coordinate is expressed as
y (x,z) = B +
The beam profiles for the x- and y-directions are given as
ω y = tω0
1 +
z
Z0
2
, ω x = uω0
1 +
z
Z0
2
where ω0 is the radius at the beam waist, Z0 is the depth of focus, and
t(0 ≤ t ≤ 1) and u(0 ≤ u ≤ 1) are variable parameters.
Next, the incident angle θ1 of a ray enteringthe sphere at the
inter-section point (x, y, z) is defined as the angle between the tangential vector
ω x , ω y , 1 of the ray and the vector b =
x, B − y (x,z) , A − z pointing
from the intersection (x, y, z) to the center (0, B, A) of the sphere
θ1= arccos ab
As a result, the trappingefficiencies Q s(x,z) and Q g(x,z) owingto a ray
hits the intersection (x, y, z) can be obtained using(3.5) and (3.6) The entire
trappingefficiency due to the entire surface of the microsphere is given later Figure 3.20 shows the sectional view of the off-axial trapping (a), indicating
how to integrate Q s(x,z) and Q g(x,z) alongthe z-axis (b) Calculate the incident angle at the arbitrary point z in the circle in the yz plane and compute the optical trappingefficiency for the ray Then integrate Q s(x,z) and Q g(x,z)along
the z-direction leadingto Qsz (x) and Qg(x) in the yz plane The integration
is carried out for the upper and lower hemispheres individually because of the dissymmetry due to off-axial trapping The integration starts from the
Beam profile (a)
A-r
zupper (x )(max)
zlower (x )(max)
Beam profile (b)
x =0
x (umax)
Fig 3.20 Method of optical pressure integration when a sphere is located at an off-axis, side view (a), and top view (b)
Trang 10100 3 Optical Tweezers
left side z (x) (tmin) = A − √ r2− x2in Fig 3.20a for both the upper and lower hemispheres The integration ends at the tangential point between the ray and the surface profiles of the upper and the lower hemispheres The integration
end points z (x)upper(tmax) for the upper hemisphere and zlower
(x) (tmax) for the lower hemisphere are given by the solution between two equations shown as
r2− x2=
y (x,z) − B 2+ (z − A)2
ω y = tω0
1 +
z
Z0
2
Then, Q z s(x) and Q z g(x)are given as
Q z s(x)=
zupper
(x) (tmax )
z (x) (tmin )
Q s(x,z) dz +
(x) (tmax )
z (x) (tmin )
Qs(x,z) dz, (3.16)
Q z g(x) =
zupper
(x) (tmax )
z (x) (tmin )
Q g(x,z) dz +
(x) (tmax )
z (x) (tmin )
Qg(x,z) dz. (3.17)
Next, our integration goes along the x-axis Figure 3.20b shows the top view, indicatinghow to integrate alongthe x-axis The trappingefficiencies
Q z
s(x) and Q z
g(x) in the yz plane are summed alongthe x-axis in the xz plane.
In this case, the integration starts from x = 0 and ends at x = x(umax), which
is the tangential point between the ray profile (3.13) and the sphere circle
(3.18) in the xz plane
x2+ (z − A)2
= r2
ω x = uω0
1 +
z
Z0
2
Then, Qall
s and Qall
g are given as
Qalls = 2
x(umax ) 0
Qallg = 2
x(umax)
0
As a result, the total trappingefficiency comes from (3.7) Followings are the numerical results for the off-axial trappingin three dimensions
Off-axial Distance and Microsphere Radius Dependence
In the analysis a circularly polarized laser beam by a laser diode with a 1.3µm wavelength, a tapered lensed optical fiber with a curvature of 10µm, and mi-crospheres 2–10µm in radius are used under the conditions listed in Table 3.6 First, transverse trappingefficiency on the off-axial distance (transverse
offset) is analyzed for a polystyrene sphere of 2.5µm radius located at different
... integrating the optical pres-sure force due to an individual ray in two dimensions On the other hand, calculation in three dimensions is necessary for the off-axial trappingeffi-ciency because of axial... whose center is located at (0, B, A) The y-coordinate is expressed asy (x,z) = B +
The beam profiles for the x- and y-directions are given as
ω... fiber with a curvature of 10µm, and mi-crospheres 2–10µm in radius are used under the conditions listed in Table 3 .6 First, transverse trappingefficiency on the off-axial distance (transverse
offset)