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3D Micromanufacturing Laboratory3D Micromanufacturing LaboratorySchool of MechatronicsSchool of Mechatronics
Gwangju Institute of Science and Technology (GIST), KOREAGwangju Institute of Science and Technology (GIST), KOREA
이 용 구
Calculation of optical trapping forces summary
Yong-Gu Lee
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Rayleigh, Mie, and Ray-optics regimes
With Rayleigh scattering, the electric field is assumed to be invariant in the vicinity of the particle
Taken from the course notes of Radar Metrology by Prof. Bob Rauber (UIUC)http://www.atmos.uiuc.edu/courses/atmos410-fa04/presentations.html
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Electromagnetic forces
+
S N
Moving + charge
Current flow direction
Electric force Magnetic force
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Electromagnetic forces
3 2 2 2
2 2 2 2
coulombs volts kilogram[ ]
meter meter second meter
amperes webers kilogram[ ]
meter meter second meter
: Electric vector
: Magnetic vector
: free charge density
: electric curr
e
V
m
V
E F E
J B F J B
E
B
J ent density
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Dielectric material (유전체 )
Dielectric material: poor conductor of electricity but an efficient supporter of electrostatic fields
Examples are: porcelain (ceramic), mica, glass, plastics, and the oxides of various metals. Dry air is an excellent dielectric. Distilled water is a fair dielectric. A vacuum is an exceptionally efficient dielectric.
Metals can be thought as dielectric at their outermost shells
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Induced electric field in a dielectric object
E1
Incidentplanewave
DielectricSphere
+
-
-
+
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Potential due to dipole
DielectricSphere
22
1 1 cos, higher order terms.
4 4
: Electric permittivity
q: Electric charge
: Charge separation
q qlr
r r r
l
- charge
+ charge2r
r
,r
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Electric field inside dielectrics
2 12 1
(1) is continuous everywhere
(2) 0 across a surface bounding two dielectrics;
it is assumed that the interface of the dielectric bears no charge,
(3)
n n
E
22 1 1 1 1
1 2
3ˆ where
2E
E E E e1
2
Image from: Julius Adams Stratton, Electromagnetic theory, McGraw-Hill Book Company Inc. 1941
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Gradient force (Rayleigh regime)
1 2 1
1 2 12 2 1 2 1
2 1
1
23 2 2
1 1 0 2
32 223 2 1 1
1 1 0 2 2
The energy of this polarized sphere in the external field is
1U
2
3
2
,
1 1, 4 ,
2 2
21 1
2 2
V
grad
grad grad T T
dv
t U
mt r n E t
m
r nm mr n E I
m c m
P E
P E E
F r
F r F r r
r r
1 2/m n n
2
1
0 0 01
1 1
21 1 0
22 2 0
1
2I E
n n
n
n
r r
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Scattering force (Rayleigh regime)
Incidentplanewave
DielectricSphere
Scattered sphericalwave
2
,
/
where is the cross section for
the radiation pressure of the particles
pr Tscat
pr
C t
c n
C
S r
F r
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Calculating Cpr
Maxwell’s equation
Decouple Maxwell’s equation
through Electric Hertz vector
Introduce the spherical scattering
geometry
Solve the decoupled
Maxwell’s equation for the scattering
cross section
Green’s function
3D Micromanufacturing Lab.3D Micromanufacturing Lab.This slide is taken from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov https://www.icfo.es/courses/biophotonics2006/html/
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Mie (scattering) theory
Spherical harmonics: Waves in spherical structures
Maxwell’s equation
Decouple Maxwell’s equation through Electric &
Magnetic Hertz vector
Solve the decoupled
Maxwell’s equation for the scattering
cross section
Spherical harmonics
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Ray optics regime
Z
Y
O
PR
P
PT
PT R
PT R
2
22
2β
αα+β
β
θθ
r
r
θ-rΠ-2(θ-r)
θ
θ
r
θ-r
r
Π-(θ+r)
Angles measured
+z
+y Medium index of refraction n1
Sphere index of refraction n2
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Scattering of a single incident ray
Why is this θ?
Scattered rays make angles relative to the incident forward ray direction of Π+2θ, α, α+β,…,α+n β,…
The powers of the scattered rays arePR, PT2, PT2R,…,PT2Rn,…
Z
Y
O
PR
P
PT
PT R
PT R
2
22
2β
αα+β
β
θθ
r
r
θ-rΠ-2(θ-r)
θ
θ
r
θ-r
r
Π-(θ+r)
Angles measured
+z
+y
Medium index of refraction n1
Sphere index of refraction n2
rnn sinsin 21
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Total scattering force Force due to a single ray of power P hitting a
dielectric sphere at an angle of θ incidence with incident momentum per second of n1p/c
0
211
0
2111
sin2sin0
cos2cos
n
ny
n
nz
nRTc
Pn
c
PRnF
nRTc
Pn
c
PRn
c
PnF
Z
Y
O
PR
P
PT
PT R
PT R
2
22
2β
αα+β
β
θθ
r
r
θ-rΠ-2(θ-r)
θ
θ
r
θ-r
r
Π-(θ+r)
Angles measured
+z
+y
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Total scattering force
0
211
0
2111
sin2sin0
cos2cos
n
ny
n
nz
nRTc
Pn
c
PRnF
nRTc
Pn
c
PRn
c
PnF
ii
tot
n
nintot
eTc
PnR
c
PniR
c
PnF
eRTc
PnR
c
PniR
c
PnF
Re1
12sin2cos1
2sin2cos1
2111
0
2111
tot z yF F F i
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Total scattering force
Gradient forceScattering force
sq
gq
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Force is towards the focal point
MICROSCOPE LENS
LASER BEAM
o
F
f
a
a b
b
Fb Fa
ab
a b
f
o
F FF
a b
a
b
a
b
f o
F
F
F
a
b
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Integrate along the beam diameter
ax
w w
r
gsaxial
Qc
Pn
drreqqdwc
PnF
1
2
0 0
2
20
1 0 20
2
sincos2
tr
w w
r
gstransverse
Qc
Pn
drreqqdwc
PnF
1
2
0 0
2
20
1 0 20
2
cossin2
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Metal trapping
An electronic field attenuates e-times in the skin layer
This slide is adapted from the lecture notes of Optical Tweezers in Biology by Prof. Dmitri Petrov https://www.icfo.es/courses/biophotonics2006/html/
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
Numerical solutions We can obtain the complete electromagnetic
solution in time and space using FEM and FDTD methods.
By integrating the Maxwell stress tensor at the surface (S) of the scattering object the optical force as well as momentum can be computed.
Fig. 1. Solid and medium under an incident field.
Ref. Seung-Yong Sung and Yong-Gu Lee, "FDTD 방법을 이용한 광집게의 포획 힘 계산 ," Hankook Kwanghak Hoeji, Vol. 19, No. 1, pp 80-83, 2008 Feb (Written in Korean)
Seung-Yong Sung and Yong-Gu Lee, “Trapping of a micro-bubble by non-paraxial Gaussian beam: computation using the FDTD method,” Optics Express, Vol. 16, No. 5, pp 3463-3473, 2008
Seung-Yong Sung “Calculations of the trapping force using the FDTD method and its applications,” 2008.02 Master’s thesis, GIST
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
The total electromagnetic force on a charge particle is
.
If the sum of all the momenta of all the particles in the volume V is denoted
by , then we have
+ .
Angular momentuV
q
ddv
dt
mech
mech
F E v B
P
PE J B
m,
+V
ddv
dt mech L
r E J B
3D Micromanufacturing Lab.3D Micromanufacturing Lab.
The force that acts on a target object immersed in a dielectric medium as illustrated in Fig. 1 due to an electromagnetic field can be represented using the Maxwell’s stress tensor [9] as
2 22S V
1 1
2
dE H da dv
c dt F E n E H n H n E H .
(1)
In Eq. (1), ε and μ are permittivity and permeability, n is the outward normal unit vector from the interior of surface S in Fig. 1, c is the speed of light in vacuum. E and H will be defined below. The surface and volume integration domain S and V represent the object surface and interior in Fig. 1, da and dv are infinitesimal area and volume. When the force in Eq. (1) is averaged for a monochromatic light in the time duration of a period T=2π/λ we get,
2 22S V
2 2
S
1 1
2
1 1
2T
dE H da dv
c dt
E H daT
F E n E H n H n E H
E n E H n H n
.
(2)
The operator <> represents the time average. The last term vanished under steady-state assumption.