The 2D optical trapping ability of larger-than average-particles is compared for three different beam types: a flat-top, a Gaussian beam, and a donut shaped beam. Optical force-displacement curves are calculated in four different size regimes of particle diameters (1.5-20 μm). We find that the trapping efficiency increases linearly with ratio of particle diameter to wavelength for all three beams. As the ratio reaches a specific threshold value, the flat-top focus exhibits the largest trapping efficiency compared to the other two beam types. We experimentally demonstrate that flat-top focusing provides the largest transverse trapping efficiency for particles as large as 20 μm in diameter for our given experimental conditions. The overall trend in our experimental results follows that observed in our simulation model. The results from this study could facilitate light manipulation of large particles.
© 2014 Optical Society of America
Optical tweezers, which harness optical forces to trap and manipulate objects, was first reported several decades ago and has since found numerous applications [1–5]. Indeed, this technology has been successfully implemented in various research applications such as atom cooling , single-molecule manipulation [7, 8], particle sorting and transportation in micrometer-scale channels , and assembling of 3D artificial structures [10, 11]. The facile manipulation of particles of various shapes, sizes, and types has been a result of exploiting the various degrees-of-freedom of the trapping laser, particularly its amplitude, phase, spatial mode, wavelength and polarization [3, 12–15]. Furthermore, since its inception, it has been well-understood that for the single-beam gradient force trap, the trapping strength is proportional to the intensity gradient of an optical field, and as such, the general route to obtaining the largest intensity gradient is to focus a standard Gaussian beam as tightly as possible. However, diffraction limits how tightly an optical field can be focused. In addition, trapping at the extremes of particles sizes, i.e., either very small or large particle diameters with respect to the optical wavelength, is challenging. Small particles undergo Brownian motion and large particles more greatly experience the effects of gravity. While various techniques have been developed over the years to improve the trapping efficiency of small particles much less has been done with respect to large particles. The ability to trap and manipulate objects that are tens of microns in diameter, with low input power, is useful in handling biological structures without damaging them. Thus, an interesting avenue to pursue with respect to trapping large particles is to use non-standard beam shapes such as a beam with a flat-top intensity distribution. Such a beam is characterized by a uniform intensity distribution with a sharp intensity roll-off along its edges and can be derived from the super-Gaussian (SG) function , which forms a continuous transition from a Gaussian to a flat-top. At the extreme case of SG, it tends to a rectangular function with an infinite gradient at both edges.
In the present work, we explore the effect of flat-top focusing on the 2D trapping of larger-than-average-particles (LAPS) with diameters in the range of 1-20 μm. Specifically, using a lens of numerical aperture (NA) of ~0.745, we employ moderate focusing of a radially polarized beam to generate a flat-top focus, an approach that we recently demonstrated . We compare the results of using a flat-top focus for trapping of LAPS to that of using a beam with a donut intensity distribution, generated from azimuthally polarized light, and to that of using the standard linearly polarized Gaussian beam. Using a three-dimensional (3D) finite element method (FEM) approach, we calculate the Maxwell stress tensor in order to model the force-displacement curves for these beams for the aforementioned particle sizes. We find that as the ratio of particle diameter to wavelength α reaches a specific threshold value, the flat-top focus exhibits the largest trapping efficiency compared to the other two beam types. In the case of our specific experimental parameters, we determine that the flat-top focus demonstrates the best trapping efficiency when trapping particles larger than 13.5 μm. We find that the overall trend in our experimental results follows that observed in simulation.
2. Computational methods
To calculate the force that acts on arbitrarily shaped micro and nanoparticles the computational procedure consists of two steps. First, the electromagnetic field distribution along the surface of the object must be solved. Next, Maxwell stress tensor analysis is applied [18, 19].
In this study, we simulate the electromagnetic field in the vicinity of focus for a standard linearly polarized Gaussian beam, a radially polarized beam, and an azimuthally polarized beams using commercial 3D FEM software (COMSOL Multiphysics v4.3b) [20, 21]. We have recently reported the intensity profile for these beams elsewhere . We analyze the optical trapping ability of these beams when used to manipulate spherical polystyrene particles of diameters 1.5, 5, 10, and 20 μm. The simulation region is a 3D cylinder and the dimensions dependent on the diameter of the particle. For the four particle sizes used, the aspect ratio (height to radius) of the cylinder is 1.2, 1.5, 1.6 and 1.8 for the 1.5, 5, 10, and 20-μm diameter particles, respectively. The grid size of the sampled incident field in the simulation volume is dependent on the predefined mesh size which is set as λ/20, where λ is the wavelength of light. For the FEM simulation, the biggest advantage is the option to apply a non-uniform mesh size in order to be computationally efficient. Thus, in our approach, we apply a finer mesh for the region of interest and a coarser mesh elsewhere. We set the global maximum element size as λ/5, and the minimum element size at the surface of a particle as λ/20. In addition, a perfect matched layer is set at all surrounding boundaries . Finally, Maxwell’s stress tensor is introduced to calculate the forces on the particles.
3. Experimental methods
Figure 1 shows a schematic diagram of the optical trapping setup. A spectrally tunable, 3W, Ti:Sapphire laser source (Spectra-Physics Mai-Tai HP DeepSee) that produces 100 femtosecond-duration pulses at 80-MHz repetition rate is operated in pseudo-CW mode . The output, linearly polarized, beam from the source is spectrally centered at 800 nm and spatially filtered before passing through a linear polarizer. Next, the beam is sent to a flip mirror which, when in the up position, directs the beam via gold mirrors and a 50:50 beamsplitter to the input of an upright microscope (Olympus IX81) with a 60 × [plan apochromatic, numerical aperture (NA) = 0.8] oil-immersion objective. When the flip mirror is in the down position, the linearly polarized Gaussian beam is first converted to a radially polarized vector beam using a radial polarization converter (Arcopix) before being directed into the microscope. We have recently demonstrated that the focusing a radially polarized beam by a lens of ~0.745 NA will produce a flat-top intensity distribution at the focus. To generate an azimuthal vector beam, a polarization rotator consisting of two half-wave plates with an angle of 45° with respect to each other, is placed between the beamsplitter and the radial polarizer . A halogen lamp, white light source (Dolan Jenner, 190) is used to image the particles (Duke Scientific) onto a CCD camera (Watec, 902H3-Ultimate), which is preceded by a laser-blocking band-pass filter (Brightline FF01-680/SP-25). For all beams, the power at the output objective is kept constant at 5 mW. The input beam waist diameter is fixed to be ~4 mm. Note that since the generated flat-top and donut beams at the focus are non-Gaussian, the standard definition of beam waist does not apply here. Trapping chambers are made with a 13-mm diameter gasket (Invitrogen CoverWell) sandwiched between two microscope coverslips (Corning). All particles used in the experiment are suspended in water and a typical concentration of 1:1000 is used. The sample is placed on a motorized stage driven by a picomotor linear actuator (Newport 8303) with 30-nm resolution.
The ability of an optical trap to convert incoming laser light (power) into a useful trapping force is measured using the dimensionless trapping efficiency parameter ,25, 26]. Experimentally, Q is determined by vibrating the trapped particle with progressively increasing speed in the transverse plane until it is ejected from the trap. These velocities are used to find the applied viscous drag force from which Q is calculated. This entire process is repeated 10 times for each input beam.
4. Results and discussion
Figure 2 shows the simulated transverse optical forces exerted on polystyrene particles of diameters 1.5, 5, 10, and 20 μm when illuminated by Gaussian, flat-top, and donut beams. As shown in Fig. 2(a), the maximum trapping force exerted by a flat-top is smaller than both Gaussian and donut shaped focus when the particle size is 1.5 μm. Overall, we observe that the ratio β of maximum transverse trapping force for the flat-top compared to the Gaussian profile is ~0.63, 0.86, 0.99, and 1.03 for the 1.5, 5, 10 and 20-μm diameter particles, respectively, as shown in the insets of Figs. 2(a)-2(d). Note that the abrupt change observed in the transverse force curve for the 10-μm and 20-μm diameter particles is a result of the relatively coarse sampling (step sizes of 500 nm) in the simulation at this particle size scale; this can result in less resolution in determining the exact maximum value of the trapping force.
Experimental results (blue bars) of transverse trapping efficiency are plotted in Fig. 3(a-d), and overall trend in our experimental results follows the trend observed in the simulation results (red bars). Note that the error in all cases is ~2% except for the case of the 1.5-μm diameter particle illuminated by a flat-top focus (the error is ~6.4%). Figure 3(e) shows the trapping efficiency using flat-top, Gaussian and donut shaped focus as a function of α. We observe from the plot that the trapping efficiency scales linearly with α. This observed linear behavior in Q is consistent with what has been reported previously for the particular range of particle diameters used in this study . In addition, the flat-top focus is seen to have the largest slope. As a result, below a critical value of α, the flat-top focus has the lowest trapping efficiency for the smallest diameter particles. In contrast, above the critical value, the trapping efficiency is largest for the flat-top focus. For the specific wavelength, numerical aperture, and spot size used in our experiments, we find that this critical value corresponds to a particle with diameter of 13.5 μm. It is worth noting that particles are trapped in 2D in the focal plane in our experiment. As the NA is 0.745 in our case, the axial gradient force is smaller than the scattering force, and thus cannot stably trap the particle axially. Thus, by pushing the particle against the coverslip, we restrict its motion to the transverse plane.
In the above experiments, we experimentally observed different trapping efficiencies for our various beam types. These results are consistent with numerical predictions. To help understand our results, it is useful to consider Fig. 4, which depicts the trapping behavior of the different beams used in our experiments for small and large particles. In the figure, the dashed arrows refer to the standard reaction force components arising from the intensity gradient, while the solid arrows are the corresponding resultant (net) force vectors. Note that the black arrows correspond to the net force exerted on the particle at the equilibrium position in the transverse plane and the green arrows correspond to the situation where the particle is off-axis. The optical trapping behavior of small particles is shown in Fig. 4 (a-c). In Fig. 4(a), the focused spot is Gaussian and the particle tends to be confined at the center of the focus as shown. If the particle is located at the center of the beam, then individual rays of light refract through the particle symmetrically, resulting in no net transverse force. When the particle is displaced from the beam center [right image in Fig. 4(a)], the larger momentum change of the more intense rays cause a net force to be applied back toward the center of the laser. When the particle is transversely centered on the beam, the resulting transverse force is zero. However, for a flat-top focus, the intensity gradient is equal to zero around the center of focus, which means that there is no transverse trapping force within the flat region of the beam (with uniform intensity distribution). This is illustrated in Fig. 4(b) where one sees only the scattering force. In this case, the transverse trapping force exists only at the edge of the flat-top region, which has a sharp roll-off. Thus, for a particle much smaller than the width of the flat-top focus, Brownian motion confines the trajectory of the particle to the transverse dimension within the flat region of intensity; as the particle approaches the edge, a restoring force pulls the particle back to the flat region. Therefore, for small particles, the flat-top focus acts as a 2D optical cage. For the donut beam, the particle is pushed away transversely to the ring-shaped region, as shown in Fig. 4(c). Next we consider the case of large particles () with diameter exceeding the width of the focused beam. The transverse equilibrium position is located at the center for all beam cases. When a large particle is off-axis, all the laser rays are refracted by the particle and the resulting momentum change causes a restoring force that pushes the particle back to the equilibrium position (geometric center). For the Gaussian beam, the intensity is mainly confined to the center. In this case, the incident light is close to a normal incidence and refraction is severely reduced, thereby leading to less momentum change, as shown in Fig. 4(d). The flat-top focus sees more of the curved region of the particle than the Gaussian beam. Thus, a larger amount of output rays are refracted and bended at the surface of the particle leading to greater momentum transfer, as shown in Fig. 4(e). This scenario explains the larger trapping efficiency of the flat-top focus when used on large particles. The donut shaped focus sees less of the curved region and the trapping force is smaller than the flat-top focus but comparable to the Gaussian beam, as we observe in Fig. 4(f).
Optical force-displacement curves of flat-top, Gaussian, and donut shaped focus for different particle sizes (1.5-20 μm) are obtained by calculating the Maxwell stress tensor over the surface enclosing the particle. The transverse optical trapping efficiency was observed both experimentally and theoretically for comparison. We found that the trapping efficiency increased linearly with ratio of particle diameter to wavelength for all beams, with the efficiency for the flat-top focus becoming comparable to the Gaussian by a certain threshold value. In the case of our specific experimental parameters, we determined that this threshold corresponded to a particle with a diameter of 13.5 μm. In addition, we experimentally verified that the transverse trapping efficiency for the flat-top focus becomes larger than that of the Gaussian beam when the particle size approaches 20 μm. The overall trend in our experimental results follows the trend observed in our simulations. In addition, we note that although the trapping by flat-top focus that we presented is based on 2D optical trapping, we anticipate that the associated larger transverse gradient force and axial scattering force makes it useful for optical levitation studies.
This work was supported by the University of Illinois at Urbana-Champaign (UIUC) research start-up funds.
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