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**Theories in optical tweezers**

2010년 2월 5일 금 14시~14시 50분 2010 겨울 광집게의 소개 및 시연회 , 광주과학기술원 기전공학과 228호

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Ashkin’s invention )

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**Three of the earliest geometries for optical tweezers**

Arthur Ashkin, Proc. Natl. Acad. Sci. 94, 4853

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**Direct manipulations Materials Shapes**

Dielectric materials 5 nm ~ 100μm Metals 5~100 nm Shapes Symmetrical shapes Sphere Rod Irregular shapes

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마이크로 테트리스

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

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**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), , 2006

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**Scattering and gradient forces**

Part 1 Scattering and gradient forces

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**Electromagnetic forces**

Electric force Magnetic force +

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광자 입자의 굴절로 인한 선형 운동량의 변화

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구형체의 포획(집기)

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**Scattering and gradient forces in optical tweezers**

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

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**Induced electric field in a dielectric object**

- E1 + - Dielectric Sphere Incident plane wave +

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**Potential due to dipole**

Dielectric Sphere + charge - charge

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**Electric field inside dielectrics**

2 1 Image from: Julius Adams Stratton, Electromagnetic theory, McGraw-Hill Book Company Inc. 1941

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**Gradient force (Rayleigh regime)**

See pp. 176 Stratton

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**Scattering force (Rayleigh regime)**

Dielectric Sphere Incident plane wave Scattered spherical wave

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**Calculating Cpr(= Cscat)**

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**Scattering force (Rayleigh regime)**

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**Numerical force calculations**

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 using the FDTD method,” Optics Express, Vol. 16, No. 5, pp , 2008

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**Position sensing of microscopic beads**

Part 2 Position sensing of microscopic beads

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Physical nature The position signal measured with a QPD in the BFP is determined by the interference of the unscattered laser beam with the scattered 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 (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)

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**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 nm and attoseconds. (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

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**Instrumentation layout**

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**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 Optical tweezers has a pN range force Optical tweezers’ force can be thought as a spring force This makes the optical stiffness (ktrap) 5μm polystyerene bead

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Sx, Sy, Sz signals 1 QPD(Quadrant Photo Diode) A,B,C,D signal로부터 x, y, z 축방향으로의 변위 signal(Sx, Sy, Sz) 계산

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**Calibration factor Laser scanning analysis**

커버글래스에 고정(stuck)된 마이크로 비드 이용 비드에 조사하는 레이저의 위치를 변화 QPD에 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

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**Force sensing (trap stiffness calibration)**

Part 3 Force sensing (trap stiffness calibration) Ref. Bechhoefer J and Wilson S Faster, cheaper, safer optical tweezers for the undergraduate laboratory. Am. J. Phys. 70:

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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 conventionally, the fluid can be moved past the particle (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 viscous 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 ascertained. Escape forces are generally somewhat different in the x,y,z directions, so the exact escape path must be determined for precise measurements. This calibration method does not require a position detector with nanometer resolution.

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Drag Force Method By applying a known viscous drag force, F, and measuring the displacement 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 triangle waves of displacement (corresponding to a square wave of force) or sine waves of displacement (corresponding 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.

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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 knowledge of the viscous drag coefficient is not required (and therefore of the particle’s geometry as well as the fluid viscosity). A fast, wellcalibrated position detector is essential, precluding video-based schemes.

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Step Response Method The trap stiffness may also be determined 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 damping time of the particle

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**Trap stiffness measurement**

1 Trap stiffness measurement Power spectrum method QPD는 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 QPD is used to recode the position of the trapped bead The power spectrum of trapped bead is fitted to the Lorentzian curve to deduce roll-off frequency

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**Comparison of methods Viscosity Geometry CCD Thermometer Piezo Stage**

QPD Remarks Escape force method T Nonlinear region Drag force method Equipartition method Linear region, Need to know Volt,displacement relations (calibration is necessary) Power spectrum method Linear region, No calibration necessary. Step response method Linear region

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**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 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 (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 , 2009 Aug

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**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

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**Contributors Thank you http://nsl.gist.ac.kr Jong-ho Baek Sun-Uk Hwang**

Seung-Yong Sung Je-Hoon Song Song-Woo Lee In-Yong Park Muhammad Tallal Bin Najam Irfan Shabbir Park Yun Hui Jung-Dae Kim Thank you

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질문/응답 -감사합니다- -수고 많으셨습니다-

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