3D Micromanufacturing Laboratory School of Mechatronics Gwangju Institute of Science and Technology (GIST), KOREA 이 용 구이 용 구 Calculation of optical trapping

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


Electromagnetic forces

+

S N

Moving + charge

Current flow direction

Electric force Magnetic force


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


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


Induced electric field in a dielectric object

E1

Incidentplanewave

DielectricSphere

+

-

-

+


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


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


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


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


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/


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


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


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


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


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


Total scattering force

Gradient forceScattering force

sq

gq


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


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


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/


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


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


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.

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3D Micromanufacturing Laboratory School of Mechatronics Gwangju Institute of Science and Technology (GIST), KOREA 이 용 구이 용 구 Calculation of optical trapping