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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
)
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory 5
마이크로 테트리스
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory 6
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory 7
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
SCATTERING AND GRADI-ENT FORCES
Part 1
8
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory 9
Electromagnetic forces
+
S N
Moving + charge
Current flow direction
Electric force Magnetic force
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광자 입자의 굴절로 인한 선형 운동량의 변화
1n
21 nn
a b
bF
2n 2n
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory 11
구형체의 포획 ( 집기 )
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory 13
Induced electric field in a di-electric object
E1
Incidentplanewave
DielectricSphere
+
-
-
+
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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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory 15
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory 16
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
POSITION SENSING OF MICROSCOPIC BEADS
Part 2
21
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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)
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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)
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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.
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory 35
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
나노 시뮬레이션 연구실Nanoscale Simulations Laboratory나노 시뮬레이션 연구실Nanoscale Simulations Laboratory 36
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
광주과학기술원([email protected])
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