1. Introduction

Single-beam optical trapping, which was first demonstrated by Ashkin in 1986 [1], is currently used for non-contact particle manipulation in a wide variety of scientific fields, including physics, chemistry, and biology. Most theoretical and experimental studies on the trapping efficiencies of micrometer- and nanometer-sized particles have considered only the intensity profile of focused laser beams [2–4]. Recently, however, numerical calculations have predicted that the polarization of a focused beam affects the trapping force on a microscopic particle [5–8]. A radially polarized beam, which is a typical cylindrical vector beam [9], is predicted to have a higher axial trapping efficiency than a linearly polarized beam. This enhanced trapping efficiency is due to all the rays of a radially polarized beam being p polarized if the center of the small particle is on the beam axis. By contrast, an azimuthally polarized beam, which is orthogonal to the radial polarization, will have a lower axial trapping efficiency than a linearly polarized beam because the beam is s polarized. The difference between these polarizations is due to Fresnel reflection on the particle surface, particularly when the particle is larger than the laser light wavelength (i.e., the ray optics regime). In fact, numerical calculations based on ray optics [5] and the T-matrix method [6,7] predict that a radially polarized beam will provide more efficient axial trapping of transparent dielectric particles than a linearly polarized beam. Most recently, Michihata et al. have reported a larger axial trapping efficiency of radial polarization than that of linear one for the optical trapping of a glass bead with a diameter of 8 μm in air [10]. In this regard, however, since the radially polarized beam used in the measurement was converted from a linearly polarized Gaussian beam by using a segmented half-wave plate, the beam was not a complete doughnut-shape with radial polarization. It has been demonstrated that the difference between Gaussian and doughnut-shaped profiles of the incident beams should be considered for the estimation of the trapping force [11]. In addition, the effect of polarization on trapping efficiency has not been experimentally verified in water, which is the most widely employed condition in the optical trapping applications.

In this paper, we demonstrate single-beam optical trapping of micrometer-sized dielectric particles by radially and azimuthally polarized beams. The effect of polarization on the axial and transverse trapping efficiencies was experimentally measured in water.

2. Experimental setup

Figure 1 shows a schematic diagram of the optical trapping setup. A radially polarized beam was generated directly from a Nd:YAG laser cavity by inserting a c-cut YVO₄ crystal in the cavity [12]. An azimuthally polarized beam was obtained by converting a radially polarized beam using a pair of half-wave plates [Fig. 2(a) ]. In addition, a doughnut-shaped linearly polarized beam, namely a Laguerre-Gaussian LG₀₁ mode, was obtained by passing an azimuthally polarized beam through a quarter-wave plate and a linear polarizer [Fig. 2(b)]. Figures 2(c)-(e) show the measured intensity distributions before the focusing lens for the radially, azimuthally, and linearly polarized beams, respectively. These figures reveal that quite similar doughnut-shaped incident beams with different polarization states were obtained by polarization conversion. Therefore, only the polarization state of the focused laser beam can produce a different trapping force in the measurements. These beams were coupled to an upright microscope with a water-immersion objective (Carl Zeiss, Plan-Neofluar 63x, NA = 1.2). Particles were suspended in water. To measure the transverse force, the motorized stage of the microscope was controlled by a PC, as described below.

 

Fig. 1 Schematic diagram of the optical trapping setup

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Fig. 2 Polarization conversion of (a) radial into azimuthal polarization using two half-wave plates and (b) azimuthal into linear polarization using a quarter-wave plate and a linear polarizer. (c)−(e) show the measured intensity distributions of radially, azimuthally, and linearly polarized beams, respectively.

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The axial trapping force along the propagation direction of a focused beam was estimated from the minimum laser power required to hold a particle three-dimensionally against gravity when the buoyant force was taken into account [13]. The axial trapping force F_ax can then be expressed as F_ax = π (ρ_b – ρ_m)D³g / 6, where ρ_b and ρ_m are the densities of the particle and the medium respectively, D is the particle diameter, and g is the gravitational acceleration. To clearly detect a particle escaping from a trap, the particles should be large and dense enough to fall quickly in the water when the laser power is insufficient to hold the particle. Accordingly, glass beads (Duke Scientific Corp.) with nominal diameters of 5 and 10 μm were used in this experiment. Five measurements were performed for each bead and with each of the three beams (i.e., the radially, azimuthally, and linearly polarized beams). Particles with different diameters were then trapped and measured in the same manner.

The transverse (i.e., perpendicular to the axial direction) trapping force was estimated from the Stokes’ drag force exerted on a particle in the medium, which was set on the motorized stage that moved perpendicular to the beam axis. The transverse trapping force F_tr is expressed as F_tr = 3πηDv_c, where η is the medium viscosity and v_c is the stage velocity when the sphere completely escapes from the optical trap. Polystyrene beads (Duke Scientific Corp.) with nominal diameters of 3, 6, and 9 μm were used for the transverse force measurements, because these particles had narrow diameter distributions. The transverse trapping forces of radially and azimuthally polarized beams were measured. Linearly polarized beams were excluded from this measurement because the trapping force varies depending on the direction of the moving stage relative to the polarization direction, and the linearly polarized beams shown in Fig. 2(e) had insufficient polarization and intensity distributions for reliable force measurement.

Figure 3 shows the trapping efficiencies calculated using parameters determined from the experimental conditions using the ray optics model [2,5]. Quantitative evaluation by the ray optics model is generally inadequate for particles whose sizes are comparable to the wavelength of the focused laser beam. However, ray optics calculations can reveal the qualitative behavior of the trapping efficiency for radial and azimuthal polarizations. Figure 3(a) shows the magnitude of the axial trapping efficiencies Q_ax for radially, azimuthally, and linearly polarized beams. The vertical axis indicates the position of the focus along the beam axis at a distance S_z from the center of the trapped particle with a radius of 1. The refractive indices of the particle and the medium are assumed to be 1.56 (glass bead) and 1.33 (water), respectively. The maximum angle of incidence is 64.5°, corresponding to the objective, which has an NA of 1.2, in water. As Fig. 3(a) clearly shows, the trapping efficiency for radial polarization is larger than those of linear and azimuthal polarizations when the focus is located near the upper surface (S_z = 1) of the particle. The maximum trapping efficiency for a radially polarized beam is approximately 1.2 times larger than that for a linearly polarized one. On the other hand, the maximum efficiency for an azimuthally polarized beam is 0.83 times smaller than that for a linearly polarized one. The transverse trapping efficiencies Q_tr for radially and azimuthally polarized beams were calculated [Fig. 3(b)]. Note that the trapping efficiency for a radially polarized beam is smaller than that for an azimuthally polarized one. The maximum trapping efficiencies for radially and azimuthally polarized beams along the transverse direction are 0.31 and 0.34, respectively.

 

Fig. 3 Calculated trapping efficiencies along (a) axial and (b) transverse directions. The circle in (a) and the quarter sector in (b) indicate the particle circumference.

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3. Results and discussion

To estimate the trapping efficiency in the axial direction, the minimum laser powers required to hold glass beads with diameters of 5 to 12 μm for radially, azimuthally, and linearly polarized beams are plotted in Fig. 4(a) . The glass bead diameters were estimated directly from microscope images [14]. Figure 4(a) shows that a radially polarized beam requires the lowest laser power for all the glass bead sizes. On the other hand, the minimum laser power required to hold a bead is the largest for an azimuthally polarized beam. A linearly polarized beam takes intermediate values between those of radial and azimuthal polarizations. The axial trapping efficiency can be derived [Fig. 4(b)] using the relation F_ax = Q_ax n P / c, where Q_ax is the trapping efficiency, n is the refractive index of the medium, P is the laser power at the focus, and c is the speed of light. For beads with diameters between 5 and 12 μm, the radial polarization has the highest axial trapping efficiency among the three polarization states. Even for radial polarization, the experimentally measured trapping efficiency was approximately 0.1 for all bead sizes, which is less than half of the calculated value. However, the experimental results are qualitatively in agreement with the calculation results shown in Fig. 3(a), because measurements of the axial trapping efficiency often tend to be smaller than those of the transverse trapping efficiency [13].

 

Fig. 4 (a) Minimum power to hold a particle as a function of a diameter of glass beads for the incident beams with radial (circles), azimuthal (squares), and linear (triangles) polarization. (b) Corresponding maximum axial trapping efficiency.

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Figure 5 shows the measured transverse trapping forces exerted on polystyrene beads with diameters of 3, 6, and 9 μm trapped by radially and azimuthally polarized beams. In all cases, the transverse trapping force of an azimuthally polarized beam was slightly larger than that of a radially polarized beam. The trapping efficiency in the transverse direction, which is given by the relation F_tr = Q_tr n P / c, can be obtained from the slope of each line in Fig. 5. The straight lines in Fig. 5 were obtained by the least squares method. For a radially polarized beam, the estimated trapping efficiencies for polystyrene beads with diameters of 3, 6, and 9 μm were 0.34, 0.40, and 0.41, respectively. On the other hand, for an azimuthally polarized beam, the trapping efficiencies for beads with diameters of 3, 6, and 9 μm were 0.36, 0.45, and 0.47, respectively.

 

Fig. 5 Measured transverse trapping force as a function of laser power for trapping polystyrene beads with nominal diameters of (a) 3 μm, (b) 6 μm, and (c) 9 μm.

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In the above experiments, we experimentally observed different trapping efficiencies for different polarization states of the beam. These results are consistent with numerical predictions [5–7]. In the ray optics regime, the higher axial trapping efficiency of radial polarization is explained by the lower reflectivity of the p polarization of a radially polarized beam. Thus, these experimental results demonstrate the advantage of using a radially polarized beam for optical trapping of micrometer-sized particles. They imply that a lower laser power can be employed to manipulate a particle or a living cell when using a radially polarized beam with than when using a linearly or circularly polarized laser beam. In addition, numerical calculations predict that a radially polarized beam is capable of trapping particles with much higher refractive indices such as semiconductors, which are difficult to trap using a conventional laser beam [5,7].

4. Conclusion

We experimentally demonstrated single-beam optical trapping of micrometer-sized dielectric particles using radially and azimuthally polarized beams. The axial trapping efficiency was measured by trapping a glass bead suspended in water. A radially polarized beam gave the highest trapping efficiency in the axial direction. On the other hand, the transverse trapping efficiency for an azimuthally polarized beam was larger than that for a radially polarized one. These results are qualitatively in agreement with calculations in the ray optics regime.