We have observed long-range trapping with a single-beam gradient force optical trap. 6 to 10 µm polystyrene beads that are initially ≈100 µm away from the trap-center can be pulled into the trap-center. Particle-tracking enables us to determine the trajectory of a bead as it moves towards the trap-center and map out a capture zone inside which trapping can occur.
©2003 Optical Society of America
The single-beam gradient force optical trap, first observed by Ashkin et al. , has found many applications in biology, ranging from force measurements on the femtonewton scale  to the study of molecular motors . The forces and properties of single-beam gradient traps such as trap stiffness have been studied extensively using micron-sized polystyrene and glass beads. To our knowledge, most of these studies have been limited to a region that is within one bead-diameter of the trap center. Both experimental  and theoretical  studies have shown that the trap exerts a maximum restoring force on a bead when the bead is displaced from the trap center by about one bead-radius. “The size of the trapping zone in a typical single-beam gradient trap is fixed and rather small, on the order of the light wavelength,”  although the interference of two annular beams has been used to create multiple trap positions and extend the size of the trapping zone . Trapping has been described as “the free floating balls literally ‘jumping’ into the trap when they pass within a ball diameter or so of the trap position.”  However, we have observed that a single-beam gradient force optical trap has a larger capture range: polystyrene beads that are initially ≈100 µm away from the trap-center can be pulled into the trap-center in a few tens of seconds.
2. Optical system
A schematic of our experimental setup is shown in Fig. 1. The source of our optical trap is a diode laser assembly (Melles Griot 56ICS115), which uses an anamorphic prism and a microlens to respectively circularize and collimate the output of an 832 nm laser diode. The laser beam enters an inverted microscope (Leica DMIRB) with bright field illumination. A Plan FLOUTAR 100× 1.3 N.A. oil immersion objective focuses the laser light into a sample chamber to form the optical trap. The 7 mm output of the diode laser assembly is expanded by a telescope consisting of a f=25-cm and a f=35-cm focal length plano-convex lens to overfill the back aperture of the microscope objective to ensure the formation of a strong trap. The sample chamber consists of a #1.5 glass microscope cover slip and a glass microscope slide separated by a layer of paraffin film. The sample chamber is filled with a colloidal suspension of polystyrene beads (Polysciences) and rests inverted on the microscope stage, with the cover slip in the bottom. The microscope objective collects light scattered by polystyrene beads in the sample chamber. After passing through a dichroic mirror and a beam-splitter internal to the microscope, the scattered light is imaged onto a video camera (GBC) and a digital still camera (Nikon Coolpix 995) via two tube lenses. A small fraction of the back-scattered laser diode light is transmitted by the dichroic mirror, which allows us to monitor our laser trap with the video camera. A digital video converter (Canopus ADVC 100) digitizes the analog video output of the still camera. A MacIntosh G4 computer records the motion of a polystyrene sphere as it is being pulled into the optical trap. The optics in the Nikon still camera does not transmit the near-infrared laser diode light, enabling us to observe the polystyrene bead without being overwhelmed by the scattered laser diode light. We calculate the laser power delivered to the sample by multiplying the input power into the back aperture of the microscope objective by the objective's transmission at 832 nm.
Figure 2 shows how we observe long-range trapping. The red lines denote the rays entering the microscope objective with the largest convergence angle ϕmax.
Once a bead has been trapped, we translate the objective to position the focal point of the objective 60 µm above the bottom of the sample chamber. By abruptly moving the microscope stage we dislodge the bead from the trap and position it away from the trap center (step 1 in Fig. 2). The slightly higher density of the bead causes it to sink, until the beam intensity is sufficient to enable trapping to occur. Long-range trapping of a 10-µm bead is recorded in the video in Fig. 3. The image of the bead is out of focus in most of the video because the bead is below the focal plane of the microscope objective. In the first 8 seconds of the video, the bead sinks and becomes more out of focus. Trapping occurs during the next 11 seconds of the video as the bead moves towards the trap-center, becoming more in focus as it approaches the focal plane. The bead is trapped in the last second of the video; it is in focus because the center of the trap is very close to the focal point of the microscope objective.
We use the image processing software ImageJ  to perform a frame-by-frame analysis of our long-range trapping videos. Performing background subtraction and grayscale-thresholding produces a black and white movie with the bead appearing as a black circle. We then use the particle-tracking routine in ImageJ to determine the x and y coordinates of the center of the bead’s image and its area in every frame of the video. The area of the bead’s image increases as the bead gets more and more out of focus. We can thus calibrate the z coordinate of the bead by finding the areas of a bead’s image at known axial positions with respect to the focal plane of the objective. When the bead is within one bead-diameter of the trap-center, diffraction produces Fresnel rings in the bead’s image, which causes the area of the bead to fluctuate as z changes. These fluctuations in the area create large uncertainties in the z coordinate of the bead when it is within one bead-diameter of the trap-center. Since we are focusing on large capture-range in this paper, this region is not particularly important. Finally, we obtain the trapping trajectory of a bead by plotting the position of the bead in each video frame. We use cylindrical coordinates to describe the position of the beads. However, due to circular symmetry around the z-axis, the θ coordinate is omitted. The trapping data we present are for polystyrene beads with diameters 2a=6 µm and 10 µm. Since 2a≫832 nm, the wavelength of the trapping laser, ray optics can be used to describe the interaction between the trapping laser and the beads. Ashkin shows that in the ray optics regime, the force exerted by a ray incident on a bead can be resolved into two components . The scattering force acts along the direction of the ray while refraction gives rise to the gradient force, which acts perpendicular to the ray.
Figure 4 displays the trajectories of two different beads at two laser powers. Each marker in a trajectory depicts the position of the bead with respect to the trap-center in one video frame. Since we record our videos at 29.97 frames/sec, adjacent markers show the movement of a bead in a 0.033 second time-interval. The dark blue trajectory in Fig. 4 shows the motion of a bead initially positioned at r=43 µm from the trap-center. Particle-tracking commences when the bead has sunk ≈20 µm below the objective focal plane. Once the bead sinks to z≈-32 µm, the beam exerts sufficient force on the bead such that it decelerates vertically and moves radially towards the beam axis (r=0). This initial radial movement is used to define the “capture zone” in the next section. After this particular bead is pulled to within 35 µm of the beam axis, it turns around vertically and follows a linear trajectory towards the beam axis. As the bead approaches the beam axis, the spacing between adjacent markers increases, which illustrates that the bead is accelerating. We estimate the acceleration of the bead to be ≈9 µm/s2 as it approaches the beam axis. This acceleration corresponds to a ≈5 pN force exerted on the bead. Once the bead reaches the beam axis, the scattering force pulls the bead quickly into the trap center. The motion of a bead in each trajectory occurs in four similar sequential steps: the bead (1) first sinks, (2) is then gradually pulled radially, primarily due to the gradient force (3) then accelerates towards the beam axis, and (4) is finally pulled rapidly towards the trap-center by the scattering force. Observing long-range trapping is more difficult at lower laser powers. If the initial radial displacement of a bead (step 1 in Fig. 2) is too large after it has been dislodged from the trap, the bead will not be able to enter a region with sufficient laser intensity to effect trapping before it sinks to the bottom of the sample chamber. Therefore, the capture-range is less for a typical trajectory at lower laser powers (e.g. the red trajectory in Fig. 4).
4. Capture zone
We then investigated the “capture zone,” the region where long-range trapping occurs. Following the initial steps described in the previous section, we vary the radial displacement r of a bead from the trap-center after it has been dislodged from the trap (step 1 in Fig. 2). As the bead sinks, we record by how far the bead sinks (i.e., the z-coordinate the bead sinks to) before trapping can occur, i.e., when the intensity of the laser beam is strong enough to start pulling the bead back towards the trap-center. The blue markers in Fig. 5 show our collection of (r,z) coordinates at which long-range trapping starts when the laser power is 29 mW.
These markers form a line that is at an angle of 64° with respect to the z-axis. We have experimentally observed that this line marks the boundary of a “capture zone” (inset in Fig. 5); if we block the trapping beam and position a bead anywhere within the capture zone, trapping will occur when we unblock the trapping beam. Since the largest convergence angle for rays entering our aqueous sample from the N.A.=1.3 microscope objective is ϕmax=78°, the trapping zone falls only within the innermost 19% of the volume of the sample chamber that is illuminated by the trapping laser.
We have demonstrated that a single-beam gradient force optical trap has a large capture-range: it can trap a polystyrene bead that is initially ≈100 µm (15 bead-diameters for 6 µm beads) away from the trap-center. However, trapping can only occur if a bead is initially positioned within the capture zone, which corresponds to a region within the conical volume that is illuminated by the trapping-laser. Further analysis of the trajectories will enable us to quantitatively understand the force acting on a bead as long-range trapping occurs.
We gratefully acknowledge J. Steven Ross for his help with implementing ImageJ in our lab, Dr. John Crocker for referring us to the Melles-Griot diode laser module, and Dr. Koen Visscher for a helpful discussion. PGS and CLB were supported by the Pomona College Summer Undergraduate Research Program.
References and links
2. J.-G. Meiners and S. R. Quake, “Femtonewton force spectroscopy of single extended DNA molecules,” Phy. Rev. Lett. 84, 5014–5017 (2000). [CrossRef]
6. K. Svoboda and S. M. Block, “Biological Applications of Optical Forces,” Annu. Rev. Biophys. Biomed. Struct. 23, 247–285 (1994). [CrossRef]
7. M. P. MacDonald, L. Paterson, K. Volke-Sepulveda, J. Arit, W. Sibbett, and K. Dholakia, “Creation and Manipulation of Three-Dimensional Optically Trapped Structures,” Science 296, 1101–1103 (2002). [CrossRef] [PubMed]
8. S. P. Smith, S. R. Bhalotra, A. L. Brody, B. L. Brown, E. K. Boyda, and Mara Prentiss, “Inexpensive optical tweezers for undergraduate laboratories,” Am. J. Phys. 67, 26–35 (1999). [CrossRef]