Manual pinzas ópticas, Resumos de Física

Baixe Manual pinzas ópticas e outras Resumos em PDF para Física, somente na Docsity!

EDU-OT 3

EDU-OT 3 /M

Portable Optical Tweezers

User Guide

Portable Optical Tweezers Kit

Portable Optical Tweezers Kit Chapter 1: Safety Rev A, April 1, 2021 Page 1

Chapter 1 Safety

1.1. Warning Symbol Definitions

Below is a list of warning symbols you may encounter in this manual or on your device. Warning: Laser Radiation

1.2. Laser Radiation Warning

Warning The class 3B laser diode used in this kit can emit more than 50 mW of optical power, which can cause damage to the eyes if viewed directly. The laser driver is equipped with a key switch and safety interlock, which should be used appropriately to avoid injury. Additionally, we recommend wearing appropriate laser safety glasses when using this kit.

Portable Optical Tweezers Kit Chapter 2: Product Description Page 2 MTN024417-D

Chapter 2 Product Description

For many people, moving and controlling objects with a beam of light sounds more like the “tractor beams” of science fiction than reality. However, optical tweezers are devices that allow precisely that kind of manipulation. Many areas of research use them to measure small forces on the order of piconewtons^1. More exotic applications include the control of tiny microgears^2. Biologists use optical tweezers to manipulate different types of molecules and cells^3. In-vitro fertilization of ova is a typical application example – sperm can be inserted into ova without mechanical contact, thus maintaining a sterile environment. In a lab course, various demonstrations and experiments can be performed with an optical tweezers setup. This kit can be used to carry out basic experiments such as moving small spheres or cells through a solution. The kit can also be used for more advanced experiments such as investigating the Brownian motion of objects, and measuring the optical forces of the tweezers. The working principle can be explained using concepts usually known to undergraduate students, such as geometric optics, basic theory of Brownian motion, and Stokes’ friction. It is an intriguing experience to be able to control objects with a laser beam – and not only for students! This Optical Tweezers Kit can be assembled into a complete and fully operating experimental setup with which particles on the order of microns can be trapped and moved. The beam path is schematically depicted in Figure 1. It is possible to perform a variety of experiments using a number of different particles such as polystyrene beads, glass beads, or starch grains from ordinary corn flour. A special feature of this setup is that it is portable. It can be moved from room to room without needing disassembly or major readjustment, making it ideally suited to demonstrate the principle of optical tweezers to students in seminars or lecture halls. Optical tweezers are not only intriguing scientific devices. Their inventor, Arthur Ashkin, also received the 2018 Nobel Prize "for the optical tweezers and their application to biological systems." As he wrote: “It is surprising that this simple [...] experiment, intended only to show simple forward motion due to laser radiation pressure, ended up demonstrating not only this force but the existence of the transverse force component [...] and stable three-dimensional particle trapping."^4 We recommend using the OTKBTK sample kit with the setup. The performance was optimized for the sample slides and the cover glasses provided with the OTKBTK. For simplicity, we designed the tweezers system to work without immersion oil. (^1) K. SVOBODA, S.M. BLOCK: Optical trapping of metallic Rayleigh particles, Optics Letters 19 (1994) 13, 930- 932 (^2) S.L. NEALE, M.P. MACDONALD, K. DHOLAKIA, T.F. KRAUSS: All-optical control of microfluidic components using form birefringence, Nature materials 4 (2005), 530- 533 (^3) J.E. MOLLOY, M.J. PADGETT: Lights, action: optical tweezers, Cont. Phys. 43 (2002) 43, 241- 258 (^4) Proc. Natl. Acad. Sci. USA, Vol. 94, pp. 4853-4860 (1997)

Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers Page 4 MTN024417-D

Chapter 3 Principles of Optical Tweezers

To describe the function of optical tweezers, we will examine the force that a focused laser beam with a Gaussian intensity profile (the TEM 00 mode) exerts on an object, which is near or in the focus. Usually one also assumes that the object is a bead, which consists of a dielectric, linear, isotropic, and spatially and chronologically non-dispersive material. In the experiments described below, micron-sized beads made of polystyrene are primarily used. It is customary to describe the force of the laser on the object by separating it into two components. One component, the scattering force, acts along the direction of beam propagation. The second component acts along the intensity gradient and is therefore called the gradient force. The gradient force can act in different directions with respect to the beam. As the laser has a Gaussian intensity profile, the gradient force can act orthogonally to the beam, but it can also act parallel to the beam, as the laser is focused and therefore also has an intensity gradient along the beam axis. These two components and their relationship to one another are the defining factors for whether or not a particle can be trapped by the optical trap. Stable optical tweezers are only obtained if the gradient force, which pulls the object in the direction of the focus, is greater than the scattering force, which pushes the particle in the direction of the beam away from the focus. The various theoretical approaches to describe optical trapping can roughly be divided according to the areas in which they are valid. The relationship of the radius 𝑅 (or diameter 𝑑) of the bead to the wavelength 𝜆 of the incident laser beam is the dividing factor. The case 𝑅 ≈ 𝜆 is theoretically very complex and shall therefore not be dealt with here. The two extreme cases for very large and very small particles are summarized below:

3.1. Dipole Approach in the Rayleigh Scattering Regime R << 

The first case we will consider is when the radius 𝑅 of the bead is significantly smaller than the wavelength 𝜆 of the incident laser beam. Then, the electrical field 𝐸⃗ (𝑟) is approximately spatially constant with respect to the particle and the situation can be portrayed as follows: As the bead is assumed to be dielectric, one can imagine it as a collection of 𝑁 point dipoles. Due to their polarizability, a dipole moment 𝑝𝑖 is induced in each of the point dipoles by the incident laser beam. Due to the linearity of the material, the following applies: 𝑝𝑖 = 𝛼 ⋅ 𝐸⃗ (𝑟𝑖)^ (1) Here, 𝑟𝑖 is the location of the i-th point dipoles and 𝐸⃗ (𝑟𝑖) is the electrical field strength at this location. In addition, the electrical field of the laser appears to be approximately spatially constant for the bead due to the condition 𝑅 ≪ 𝜆, meaning that at a certain point in time 𝑡 0 the strength of the electrical field is equally great for all point dipoles of the bead. As a result, the induced dipole moment is equally great for all 𝑁 point dipoles. The polarization 𝑃 resulting from the induced dipole moments is then

Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers Rev A, April 1, 2021 Page 5

∑ 𝑝 𝑖

⋅ 𝛼 ⋅ 𝐸⃗ = 𝜒 ⋅ 𝜖 0 ⋅ 𝐸⃗ (^) (2) where 𝜒 is the electrical susceptibility, 𝜖 0 is the electrical constant, and 𝑉 is the volume of the bead. The potential energy 𝑈𝑖 of one of the point dipoles with dipole moment 𝑝 in the electrical field 𝐸⃗ is 𝑈𝑖 = −𝑝𝐸⃗. Because there are 𝑁 point dipoles in a bead with the volume 𝑉, the energy density in the bead is defined by: 𝑈 =

⏟⋅^ 𝑝

𝑃⃗

The occurrence of the gradient force, which is a force component that is directed in the direction of the intensity gradient of the incident electrical field, can be explained when one observes this potential energy 𝑈 of the bead in the electrical field. Equation (2) states that 𝑃 is proportional to 𝐸⃗. Therefore, according to equation (3), 𝑈 is proportional to |𝐸⃗ | 2 and thus to intensity 𝐼 ∝ |𝐸⃗ | 2 of the incident field. The force exerted on the particle by the incident field is proportional to the gradient of the potential energy ∇𝑈 and therefore proportional to the intensity gradient ∇𝐼. The following equations describe the gradient force: 𝐹𝐺𝑟𝑎𝑑 =

𝑐 𝑛𝑚^2

𝛼 = 𝑛𝑚^2 𝑅^3 (

𝑚^2 − 1

𝑚^2 + 2

𝑛𝑚^ (6)

Here, 𝛼 is the polarizability of the dipoles and 𝑚 is the relationship of the refraction index of the particles, 𝑛𝑝 (polystyrene in our case) to the refraction index of the surrounding medium, 𝑛𝑚 (water in our case). The destabilizing scattering force component is explained by the scattering of the incident light at the particle. The force action is created by the absorption and isotropic re-emission of the light by the bead. As 𝑅 ≪ 𝜆, the conditions are fulfilled for Rayleigh scattering. The resulting force can be stated as follows: 𝐹𝑆𝑐𝑎𝑡𝑡𝑒𝑟𝑖𝑛𝑔 =

128 𝜋^5 𝑅^6

3 𝜆^4

𝑚^2 − 1

𝑚^2 + 2

2 (8)

Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers Rev A, April 1, 2021 Page 7 Reflection and Transmission of an Incident Partial Beam on the Inside and Outside Surfaces of a Sample Bead with a Refractive Index Higher than the Immersion Medium Inside the sphere, the beam is reflected and transmitted numerous times. Part of the beam is repeatedly reflected on the sphere's internal wall and remains in the sphere, while the rest exits the sphere again through transmission. The beams which exit the sphere again were thus subject to a change in momentum 𝑑𝑝 𝑑𝑡

. The force on the sphere is equal to the momentum per unit time that remains in the sphere, based on equation (9). Now, the force on the sphere is once again divided into two components: a component in the direction of the incident beam (corresponds to the Z direction) and a perpendicular component (corresponds to the Y axis). This results in the following for both forces: 𝐹𝑠 ≔ 𝐹𝑧 = 𝑛𝑚

with the Q factor 𝑄𝑠 = 1 + 𝑅𝑟 cos( 2 𝜃)^ − 𝑇^2 (cos( 2 𝜃 − 2 𝑡) + 𝑅𝑟 cos( 2 𝜃)) 1 + 𝑅𝑟^2 + 2 𝑅𝑟 cos( 2 𝑡)^

and 𝐹𝑔  𝐹𝑦 = 𝑛𝑚

with the Q factor

Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers Page 8 MTN024417-D 𝑄𝑔 = 𝑅𝑟 sin( 2 𝜃) − 𝑇^2 (sin( 2 𝜃 − 2 𝑡) + 𝑅𝑟 sin( 2 𝜃)) 1 + 𝑅𝑟^2 + 2 𝑅𝑟 cos( 2 𝑡)^

Here, 𝑡 is the angle at which the first transmitted beam is refracted toward the normal (see Figure 2). According to Snell’s law of refraction, the following relationship is in effect for the angles 𝜃 and 𝑡: sin(𝜃) sin(𝑡)

𝑄𝑠 and 𝑄𝑔 are dimensionless Q factors, which state what percentage of the incident momentum contributes to the force parallel or perpendicular to the beam, respectively. These factors depend heavily on the angle of incidence of the beam, as one can see from the equations. This angle becomes larger the more heavily the beam is focused, which occurs when a higher numerical aperture objective is used. The component of the beam that points in the incident direction (Z direction) ultimately causes the scattering force 𝐹𝑠. The component perpendicular to this (Y direction) is mainly responsible for the gradient force 𝐹𝑔. In order to obtain the overall power, one must naturally consider all partial beams and integrate all of them. That will be discussed in detail below. Q Factor Angular Dependence^5 Figure 3 shows the values of the two 𝑄 −factors, depending upon the angle of incidence 𝜃 when the focus is located slightly above the surface of the sphere. One can see here 𝑄𝑔 is negative through almost the entire range, meaning the force acts in the negative Y

Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers Page 10 MTN024417-D So far, the force was given as a function of 𝜃 which now has to be expressed in terms of 𝑟 and 𝛽. Figure 5 shows a partial beam incident on the sphere with angle 𝜃 from the side. The following statements hold true: 𝑋 = sin 𝜙 ∙ 𝑆 = sin 𝜃 ∙ 𝑅 (1^9 ) and sin 𝜙 = 𝑟 ⁄√ 𝑟^2 + 𝑙^2 (^20 ) 𝑙 roughly corresponds to the focal length of the objective. Effectively, you can use the objective’s working distance for this parameter. Relation Between 𝒓 and 𝜽   Then, 𝜃 can be expressed as 𝜃(𝑟) = arcsin (

√𝑟^2 + 𝑙^2

Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers Rev A, April 1, 2021 Page 11 Next, the forces need to be summed. For that, we start by observing Figure 6. A partial beam with distance 𝑟 to the symmetry axis of the whole beam falls on the sphere under an angle 𝜃. As discussed above, we can split the resulting force in two perpendicular components, 𝐹𝑠 and 𝐹𝑔. For clarity, we can now add another partial beam, namely the one mirrored on the symmetry axis, denoted “mirror beam”. As sketched in Figure 6, this partial beam falls on the sphere on the other side and results in “mirrored” force vectors 𝐹𝑠,𝑚𝑖𝑟𝑟𝑜𝑟 and 𝐹𝑔,𝑚𝑖𝑟𝑟𝑜𝑟 in the right part of the sphere (which were not drawn to avoid an overcrowded figure). The Contribution of One Ray to the Total Force^5 ,^6 When you consider all of the partial beams with the same condition around the symmetry axis, it immediately becomes clear that all force components in X and Y direction vanish and only resulting force components along the Z axis remain. These can be written as 𝐹𝑔,𝑧 = −𝐹𝑔 ∙ sin 𝜙 = −𝐹𝑔 ∙

√𝑟^2 + 𝑙^2 (22)

𝐹𝑠,𝑧 = 𝐹𝑠 ∙ cos 𝜙 = 𝐹𝑠 ∙

√𝑟^2 + 𝑙^2 (23)

where 𝐹𝑔,𝑧 is pointing in the negative Z direction and is, therefore, negative. Hence, each infinitesimal force contributing to the total force is given by 𝑑𝐹 = 𝐹𝑠 ∙ cos 𝜙 − 𝐹𝑔 ∙ sin 𝜙 (^) (24) (^6) Adapted from A. Langendörfer: ”Aufbau einer Optischen Pinzette für das Landesmuseum für Technik und Arbeit in Mannheim“, wissenshaftliche Arbeit, KIT, Karlsruhe, 2009

Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers Rev A, April 1, 2021 Page 13

3. When the focus is below the sphere’s center, i.e. 𝑆/𝑅 ≤ 0 , scattering and gradient force act in the same direction. When the focus is above the sphere’s center, i.e. 𝑆/𝑅 ≥ 0 , both forces point in opposite directions. We also note that the absolute value of the gradient force is always higher than the absolute value of the scattering force. This is the requirement for any stable optical tweezers trap. The fundamental quantity to fulfill this requirement is the microscope objective’s numerical aperture, which defines the angle of the focus which we will discuss next. Sometimes the focus is not above or below the sphere’s center but along the y-axis instead. Again, the reference axis goes through the sphere’s center, and we plot the occurring forces as a function of the focus’ distance 𝑆 to the sphere’s center in units of the sphere’s radius. Figure 8 shows the gradient force and the scattering force. Plotting the sum of both forces would not make sense as they point in different directions (𝐹𝑠,𝑡𝑜𝑡 in Z direction and 𝐹𝑔,𝑡𝑜𝑡 in Y direction). From Figure 8, we learn that:
4. The general form of the curves is similar to the focus’ movement on the Z axis. Only outside of the sphere the force decreases a little faster than compared to the Z axis.
5. The maximal gradient force is larger than the maximal scattering force (absolute values). This is the case when the parameters such as the numerical aperture allow optical trapping.
6. The maximal gradient force are located just within the sphere, close to the surface. The maximal value of the scattering force is found right on the surface. Scattering and Gradient Force with the Focus Changed in Y Direction^6 Gradient force, 𝐹𝑔,𝑡𝑜𝑡 Scattering force, 𝐹𝑠,𝑡𝑜𝑡 Force [a.u.] Location of the focus in units of the sphere radius 𝑅 (Y direction)

Portable Optical Tweezers Kit Chapter 3: Principles of Optical Tweezers Page 14 MTN024417-D Influence of the Numerical Aperture As we have seen above, the angle of incidence of the partial rays plays a crucial role in optical tweezing. The angle is defined by the numerical aperture of the objective: the numerical aperture (𝑁𝐴) describes the acceptance cone of an objective and is given by 𝑁𝐴 = 𝑛 ∙ sin 𝜙 (^) (27) where 𝑛 is the refractive index of the material between the objective and the focus and 𝜙 is half of the angle of the maximum light cone. As discussed, the gradient force 𝐹𝑔,𝑡𝑜𝑡 needs to exceed the scattering force 𝐹𝑠,𝑡𝑜𝑡 to get a stable trap. Next, we want to investigate a measure for the trap’s strength. For that, we can have a look at the ratio of 𝐹𝑔,𝑡𝑜𝑡 and 𝐹𝑠,𝑡𝑜𝑡 at the point 𝑆/𝑅 = 1 since we have shown that scattering and gradient force are strongest when the laser focus is at/near the sphere’s surface (i.e., |𝑆 ⁄𝑅 | = 1 ). So for a stable trap, we can assume the condition^7 : |

𝐹𝑠,𝑔𝑒𝑠(𝑆 ⁄^ 𝑅= 1 )

Absolute values are used since 𝐹𝑔,𝑡𝑜𝑡 is negative. Figure 9 shows how this strength depends on the numerical aperture. Again, we plotted a curve with typical parameters to show the general behavior of the curve; therefore, we do not focus on the concrete numbers. Figure 9 shows the fundamental behavior that the strength of the trap increases with increasing numerical aperture. Also, it becomes apparent that there is a lower limit for the numerical aperture of the objective. Below that, no trapping occurs since the gradient force never exceeds the scattering force. Behavior of the Trap’s Strength with Respect to the Numerical Aperture^6 (^7) The absolute value of the forces is used since 𝐹𝑔,𝑔𝑒𝑠 is negative. |𝐹 𝑔,𝑡𝑜𝑡 / 𝐹 𝑠,𝑡𝑜𝑡 | Numerical aperture 𝑁𝐴

Portable Optical Tweezers Kit Chapter 4: Kit Components Page 16 MTN024417-D

Chapter 4 Kit Components

In cases where the metric and imperial kits contain parts with different item numbers, metric part numbers and measurements are indicated by parentheses unless otherwise noted.

4.1. Trapping Laser Source

1 x SR9A-DB ESD Protection and Strain Relief Cable 1 x L658P 658 nm, 40 mW, Ø5. 6 mm, A Pin Code Laser Diode 1 x LTN330-A Adjustable Collimator for Ø5.6 mm Laser Diodes, AR Coated: 350 – 700 nm 1 x KLD 101 K-Cube Laser Diode Driver 1 x TPS ±15 V / 5 V K-Cube Power Supply 1 x^ RS3.5P8E ( RS3.5P4M) Ø1" (Ø25 mm) Pedestal Post, 3.5" (90 mm) Tall 1 x CF Small Clamping Fork (^) Ø1"1 x Cage^ KC1 - Compatible - T(/M) SM1-Threaded Mirror Mount 1 x AD15F SM1-Threaded Adapter for Ø15 mm Components

Portable Optical Tweezers Kit Chapter 4: Kit Components Rev A, April 1, 2021 Page 17

4.2. Beam Expander

2 x ER 10 Ø6 mm Cage Assembly Rod, 10" Long 2 x ER 1 Ø6 mm Cage Assembly Rod, 1" Long 2 x ER 3 Ø6 mm Cage Assembly Rod, 3" Long 2 x ER 6 Ø6 mm Cage Assembly Rod, 6" Long 2 x^ CP^45 (/M) 30 mm Removable Segment Cage Plate 2 x CP45T(/M) 30 mm Removable Segment Cage Plate, Thick 1 x LA1509-A Ø1" N-BK7 Plano-Convex Lens, f = 100 mm AR Coating: 350-700 nm 1 x SM1A Adapter with External SM1 Threads and Internal SM05 Threads 1 x SM05L Ø1/2" Lens Tube, 0.3" Long 1 x LA1074-A Ø1/2" N-BK7 Plano- Convex Lens, f = 20 mm, AR Coating: 350-700 nm 1 x^ TR3 (TR75/M) Ø1/2" (Ø12.7 mm) Post, 3" (75 mm) Long 1 x PH3 (PH75/M) Ø1/2" (Ø12.7 mm) Post Holder, 3" (75 mm) Long