광주과학기술원 ([email protected]) 이 용 구이 용 구 1 theories in optical tweezers 2010...

광주과학기술원([email protected])

이 용 구

1

Theories in optical tweezers

2010 겨울 광집게의 소개 및 시연회 2010.02.05, 광주과학기술원 기전공학과 228 호

2010 년 2 월 5 일 금 14 시 ~14 시 50 분

나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory

Ashkin’s invention

2

)


Three of the earliest geometries for optical tweezers

3

여러 경우가 가능하나 맨 왼쪽 경우만 “ optical tweezers” 라고 한다 .Arthur Ashkin, Proc. Natl. Acad. Sci. 94, 4853

나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory 4

Direct manipulations Materials

Dielectric materials 5 nm ~ 100μm Metals 5~100 nm

Shapes Symmetrical shapes

Sphere Rod

Irregular shapes


마이크로 테트리스


3.3 nm particles trapping

Lingyun Pan, Atsushi Ishikawa, and Naoto Tamai, "Detection of optical trapping of CdTe quantum dots by two-photon-induced lumi-nescence," Physical Review B Vol. 75, 161305, 2007


Indirect manipulations Handles

Graeme Whyte, Graham Gibson, Jonathan Leach, and Miles Padgett. “An optical trapped microhand for manipulating micron-sized objects,” Opt. Express 14(25), 12497-12502, 2006


SCATTERING AND GRADI-ENT FORCES

Part 1

8


Electromagnetic forces

+

S N

Moving + charge

Current flow direction

Electric force Magnetic force


광자 입자의 굴절로 인한 선형 운동량의 변화

1n

21 nn

a b

bF

2n 2n


구형체의 포획 ( 집기 )


Scattering and gradient forces in op-tical tweezers

12Ref Christine Piggee, "Optical tweezers: not just for physicists anymore" Anal. Chem. Vol. 81, pp. 16–19, 2009


Induced electric field in a di-electric 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 di-electrics

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 1 1

2 1 21 1 2 2 1

1 2

1

23 2 2

1 2 0 2

32 223 2 1 2

1 2 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

2 22 0 2 1 0 1

/

,

m n n

n n


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(= Cscat)

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


Scattering force (Rayleigh regime)

2

2

,ˆ ,

/

ˆwhere is the cross section for the radiation pressure of the particles and

is the unitvector in the beam propagation direction. In case of small dielectric

pa

pr Tscat pr

pr

C t nC I

c n c

C

S rF r z r

z

22

4 22

rticles in the Rayleigh regime where the particle scatters the light isotropically,

is equal to the scattering cross section.

8 1

3 2

pr

pr scat

C

mC C ka a

m

1 2

2 22 0 2 1 0 1

/

,

m n n

n n


Numerical force calcu-lations All analytical force solutions can not incorporate

tightly focused beams Finite Difference Time Difference method is widely

accepted because it is able to formulate arbitrary geometry and laser sources Seung-Yong Sung and Yong-Gu Lee, "Calculations of the

trapping force of optical tweezers using FDTD Method," 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 us-ing the FDTD method,” Optics Express, Vol. 16, No. 5, pp 3463-3473, 2008

20


POSITION SENSING OF MICROSCOPIC BEADS

Part 2

21


Physical nature

The position signal measured with a QPD in the BFP is determined by the interference of the unscattered laser beam with the scat-tered light.

The trapped sphere is described as a Rayleigh scatterer (i.e., a dielectric sphere with a radius a much smaller than the wavelength)

Ref. Pralle, A. et al, Three dimensional high-resolution particle tracking for optical tweezers by forward scattered light, Microscopy Research and Technique, Vol. 44, pp 378-386 (1999)Gittes, F. and C. F. Schmidt, Interference model for back-focal-plane displacement detection in optical tweezers. Optics Letters, Vol. 23, pp 7-9 (1998)


Near to far field transformation Scattering from a spherical bead by a focused Gaussian beam problem is reduced to

that by a planar wave Scattered field was computed numerically by FDTD and transformed to the far

field 304 nm diameter polystyrene sphere immersed in water using a 633 nm laser. The spatial and time resolution was 0.1583 nm and 0.2639 attoseconds.

0 18 36 54 72 90 108 126 144 162 1800

1

2

3

4

5

6

7x 10

-9

0 18 36 54 72 90 108 126 144 162 1800

20

40

60

80

100

120

(a) Scattering magnitude (b) error as a function of deflection angles (%)

Ref. Optical Society of Korea Winter Annual Meeting 2010, 이용구 , 시간영역 유한차분법에서 근접장으로부터 원격장으로의 변환


Instrumentation layout

AB

C D

AB

C D

SLM

CMOS Camera

P-pol

M1

S-pol

PBS1

L2

L1

L3 L5

Objective

Condenser

L6 ND2

QPD1

DM2

DM1

M4

L8

60x1.2 NAL4

PBS2

QPD2

L7

M2

M3

PBS3

ND1

Trapping Laser Lamp

PBS: Polarizing BeamsplitterDM: Dichroic Mirror

Dual trap

MotorizedStage


Movement of a bead under linear spring force

광집게 (Optical tweezers) 는 피코뉴턴 (pN) 단위의 포획 힘을 가짐 이때의 포획 힘은 스프링 (ktrap) 처럼 작용

Zoomed

Trapped bead F=ktrapdx

dx

Laser beam5μm polystyerene bead


Sx, Sy, Sz signals

• QPD(Quadrant Photo Diode) A,B,C,D signal 로부터 x, y, z 축방향으로의 변위 signal(Sx, Sy, Sz) 계산

[26]

0 500 1000 1500 20006.4

6.6

6.8

7

Time (ms)

Sz

(arb

. un

its)

x

y

DC

A 0 500 1000 1500 2000-0.2

0

0.2

Time (ms)

Sx

(arb

. un

its)

0 500 1000 1500 2000-0.2

0

0.2

Time (ms)S

y (a

rb.

units

)

X-Axis Difference(Sx): (A+D)-(B+C)Y-Axis Difference(Sy): (A+B)-(C+D)Z-Axis Difference(Sz): (A+B+C+D)

B

나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory Nanoscale Simulations

LabDepartment of Mecha-tronics

Calibration factor

Laser scanning analysis 커버글래스에 고정 (stuck) 된 마이크로 비드 이용 비드에 조사하는 레이저의 위치를 변화 QPD 에 측정되는 신호를 수집

스캐닝을 이용한 QPD 시그널의 변화 측정 (Calibration factor)

수집된 신호 (Sx, Sy, Sz) 에 기울기 (Calibration factor) 값을 대입하여 실제 비드의 이동거리 계산

[27]

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

x position (m)

Sx

(arb

. un

its)

Slope at the linear region

-6 -4 -2 0 2 4 6-6

-4

-2

0

2

4

6

y position (m)

Sy

(arb

. un

its)

-15 -10 -5 0 5 10 15 203

4

5

6

7

8

z position (m)

Sz

(arb

. un

its)

Slope at the linear region Slope at the linear region

x axis scan y axis scan z axis scan

B

DC

A B

DC

A B

DC

AQPD

Lens

Moving the stage x-direction

Cover glass

Stucked bead(5μm)


FORCE SENSING (TRAP STIFFNESS CALIBRATION)

Part 3

28

Ref. Bechhoefer J and Wilson S. 2002. Faster, cheaper, safer optical tweezers for the undergraduate laboratory. Am. J. Phys. 70: 393-400.


Escape force method This method determines the minimal force required

to pull an object free of the trap entirely, generally accomplished by imposing a viscous drag force whose magnitude can be computed

To produce the necessary force, the particle may either be pulled through the fluid (by moving the trap relative to a stationary stage), or more con-ventionally, the fluid can be moved past the parti-cle (by moving the stage relative to a stationary trap).

The particle velocity immediately after escape is measured from the video record, which permits an estimate of the escape force, provided that the vis-cous drag coefficient of the particle is known. While somewhat crude, this technique permits calibration of force to within about 10%.

Note that escape forces are determined by optical properties at the very edges of the trap, where the restoring force is no longer a linear function of the displacement. Since the measurement is not at the center of the trap, the trap stiffness cannot be as-certained.

Escape forces are generally somewhat different in the x,y,z directions, so the exact escape path must be determined for precise measurements. This cal-ibration method does not require a position detec-tor with nanometer resolution.

max 6

: viscosity

R: radius of particle

escapeF R V


Drag Force Method By applying a known viscous drag

force, F, and measuring the dis-placement produced from the trap center, x, the stiffness k follows from k=F/x.

In practice, drag forces are usually produced by periodic movement of the microscope stage while holding the particle in a fixed trap: either tri-angle waves of displacement (corre-sponding to a square wave of force) or sine waves of displacement (cor-responding to cosine waves of force) work well

Once trap stiffness is determined, optical forces can be computed from knowledge of the particle position relative to the trap center, provided that measurements are made within the linear (Hookeian) region of the trap. Apart from the need for a well-calibrated piezo stage and position detector, the viscous drag on the particle must be known.

QPD

CCD

6

: viscosity

R: radius of particle

F R V

Displacement

Forcek=F/D


Equipartition Method One of the simplest and most straightforward

ways of determining trap stiffness is to measure the thermal fluctuations in position of a trapped particle. The stiffness of the tweezers is then computed from the Equipartition theorem for a particle bound in a harmonic potential:

The chief advantage of this method is that knowl-edge of the viscous drag coefficient is not re-quired (and therefore of the particle’s geometry as well as the fluid viscosity). A fast, wellcali-brated position detector is essential, precluding video-based schemes.

21 1

2 2BE k T k x


Step Response Method The trap stiffness may also be de-

termined by finding the response of a particle to a rapid, stepwise movement of the trap

harder to identify extraneous sources of noise or artifact using this approach. The time constant for movement of the trap must be faster than the characteristic damp-ing time of the particle

For small steps of the trap, the response is given by below where

is trap stiffness and is the viscous drag.

1

t b

kt

b t

x x

k

x x e


Trap stiffness mea-surement• Power spectrum method

– QPD 는 포획된 비드의 위치를 측정– 푸리에 변환을 거쳐 로렌츠 (Lorentzian) 곡선으로 피팅– 로렌츠 곡선으로 부터 roll-off frequency 도출

f: frequency, kb: Boltzmann's constant, T: Temperature,

β=6πγa : hydrodynamic drag coefficient, a: radius of the particle, γ: drag coefficient

• fc : Roll-off frequency– 주파수 성분이 가지는 파워가 절반으로

떨어지는 지점– : Trap stiffness 계산

[33]

2 2 2( )

( )B

vc

k TS f

f f

fc=26.58Hz2ck f


Comparison of meth-ods

Viscosity Geome-try

CCD Ther-mometer

Piezo Stage QPD Remarks

Escape force method

T T T Nonlinear region

Drag force method

T T T T Nonlinear region

Equipartition method

T T Linear region, Need to know Volt,displacement relations (calibra-tion is necessary)

Power spec-trum method

T T T Linear region, No calibration neces-sary.

Step re-sponse method

T T T Linear region


Optical tweezers as a force measurement device and a monitor

Interference pattering detection Quadrant photo diode 10nm resolution typically 50pN/μm spring constant 100pN~1pN can be measured

Microscope Brightfield Fluroscent

QPD

Diode Laser 685 nm

CCD

Illuminator

Workstation

L5

DM4

DM2

Filter

Condenser

Objective

DM3

L6

L7

L8 L9

M3

M2

F = k x D

D

k: spring constant

Ref. Pralle, A. et al, Three dimen-sional high-reso-lution particle tracking for op-tical tweezers by forward scat-tered light, Mi-croscopy Re-search and Technique,Vol. 44, pp 378-386 (1999)

Sun-Uk Hwang and Yong-Gu Lee, "Influence of time delay on trap stiffness in computer-controlled scanning optical tweezers," Journal of Optics A: Pure and Applied Optics, Vol. 11, No. 8, pp 085303, 2009 Aug


Summary Scattering and gradient forces

Correct optical tweezers geometry Direct manipulations Indirect manipulations

Position sensing of microscopic beads QPD Calibration factor

Force sensing Power spectrum method


이 용 구

37

Contributors

Thank youhttp://nsl.gist.ac.kr

Jung-Dae Kim

Seung-Yong Sung

Jong-ho BaekSun-Uk Hwang

Song-Woo LeeIn-Yong Park

Je-Hoon Song

Muhammad Tallal Bin NajamIrfan Shabbir Park Yun Hui


이 용 구

38

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