Optical trapping with π-phase cylindrical vector beams

The use of π-phase radially and azimuthally polarized vector beams in optical trapping is investigated. We find that by tuning the relative phase between the eigenmodes comprising the beams, the optical forces applied to a trapped particle are modified. In particular, axial trapping efficiency is enhanced with increasing z polarization and the lateral trapping efficiency of the vector beams is reduced compared to a Gaussian input beam. In addition, this is the first experimental demonstration of low-power optical trapping in an aqueous environment using vector beams, which may have important applications in biological systems.


Theory
Conceptually, optical trapping concerns the competition between the gradient and scattering forces. The former is proportional to the intensity gradient of the focused optical field, and acts to pull a high-index (relative to the surrounding medium) particle into the high-intensity region of the focal field [21]. In the Mie regime, where the particle size is much larger than the optical wavelength, the scattering force can be interpreted as resulting primarily from the reflection of the s-polarized incident field from the particle surface [22,23]. While the p-polarized field does contribute somewhat, its average reflection coefficient is 50% lower than that of the s-polarized field for the angles covered by a 1.25 NA objective. In the opposite Rayleigh regime, the particle size is much smaller than the wavelength, and the scattering force is interpreted in a dipole picture, where it is proportional to the time-averaged Poynting vector [24]. Determination of the scattering force in the intermediate-size regime is more challenging and relies on rigorous EM calculations using T -matrix or finite-difference time domain (FDTD) methods [25]. In all cases, the polarization structure and spatial intensity distribution of the focused trapping beam are important parameters for determining the optical forces on trapped particles.
We can express a Hermite-Gaussian beam with polarization components along thex and y directions by E i = H nmx + H nmŷ [26] with where A 0 is the field amplitude, w 0 is the beam waist diameter and H i (u) is a Hermite polynomial with index i. In order to model the input vector beams, we need only consider the first two modes of H i given by H 0 (u) = 1 and H 1 (u) = 2u. Following the method of Richards and Wolf [26,27], we find the electric field at the focal plane of an aplanatic lens (microscope objective) via where E 0 is the refracted field just beyond the lens, A ρϕz = j k f exp(−j k f )/(2π), k is the wavenumber and f is the lens focal length, ρ and φ(ϕ) are the polar coordinates at the surface of the lens (focal plane) and θ is the coordinate obtained from refraction of ρ at the lens surface, and θ m = sin −1 (NA/n 2 ) with NA = 1.25 and n 2 = 1.518. The refracted field in (2) is obtained by converting (1) to the spherical coordinates ( f, θ, φ), which gives wherex = cos φθ − sin φφ andŷ = sin φθ + cos φφ , with the standard spherical coordinate unit vectors given byθ andφ. The additional term [(n 1 /n 2 ) cos θ ] 1/2 ensures conservation of energy across the lens [26]. Inasmuch as (3) represents a general input beam after refraction, we can express the specific beams used in 4 this study by the following formulae: wherex andŷ are defined as in (4). The integral in (2) can be simplified by making use of the following identities: where J n is a Bessel function of order n. Using (3)-(6), we can express (2) as a single integral over θ , which upon evaluation produces the electric field in the focal plane for various input beams. The focal intensities corresponding to each polarization component are given by y, z and ' * ' represents complex conjugation. We therefore find the z-component of the focal intensity for each input beam via , which represents the objective filling parameter. In this case, a w 0 corresponding to a 10% overfilling of the objective back aperture is used. Similar expressions are used to find the total focal plane intensity of each beam, defined by I tot = I x + I y + I z , but are omitted here for brevity.
Using this formalism, the z-polarized and total focal intensity distributions for the various input beams are calculated by numerical integration and are shown in figure 1. The focal intensity distributions for each input beam are normalized with respect to each other, and the scale bars give the percentage of the maximum intensity present in each beam (e.g. the π-radial  beam has 20% the value of z-polarized intensity present in the radial beam). It can be seen that the radial beam produces the largest z polarization, as expected.
The calculated focal intensity gradients for the beams are shown alongside their respective total intensity distributions in figure 2. Note that the π-azimuthal has the same profile as the π-radial, yet rotated by 45 • ; hence it is not included in the plots. Since the gradient force is proportional to the intensity gradient [21], figure 2(a) provides a visual comparison of the maximum gradient forces derived from each beam. From figure 2(b) we observe that the Gaussian and radial beams have comparable spot sizes (∼500 nm) whereas the spot sizes of the π -phase and azimuthal beams are considerably larger (∼900 and 800 nm, respectively). Note that the simulations are performed using the same conditions as stated for figure 1. It is clear that the intensity gradient is strictly negative (positive) for positive (negative) particle displacements for the radial and Gaussian input beams, which indicates that the gradient force always acts in the direction opposite to particle displacement, thereby serving to restore the particle to the trap center. Conversely, the sign changes in the intensity gradient for the π-phase and azimuthal beams imply that the gradient force (i) is only large over small regions in the focal plane, and (ii) does not always act opposite to the displacement, i.e. it sometimes acts to reject the particle.

Experiment
The experimental setup is shown in figure 3(a). A Mach-Zehnder interferometer equipped with tunable phase plates is used to create the vector beams, and the input beam derives from a 100 mW, 660 nm wavelength diode laser (Newport, LQC660-110C). The output of the interferometer is spatially filtered, expanded and coupled into a custom-built microscope, which consists of a 100× (1.25 NA) objective (Edmund Optics-NT43-905) and a 40× (0.65 NA) condenser (Newport-MV-40×). A white light source (Dolan Jenner, 190) is used to image the 1 µm 7 polystyrene particles (Duke Scientific) onto a CCD camera (Watec, 902H3-Ultimate). Using a flip mirror the output beam can be selectively diverted into the polarization analyzer arm, from which the purity of the vector beam polarization profile can be checked. In addition, the output can be switched to a Gaussian beam by blocking one arm of the interferometer and removing the phase plate from the remaining arm. For all beams, the power at the input to the microscope is kept constant at 3 mW. To tune the relative phase between the two eigenmodes to π, one simply extends the optical path length of a single phase plate. The various vector beams discussed previously are shown in figure 3(b).
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 where F max is the maximum trapping force exerted on the particle, c is the speed of light, n is the refractive index of the surrounding medium and P is the input laser power [22]. The trapping efficiency is assessed via the Stokes drag-force method in which we drive the sample stage at velocity ν, inducing a drag force on the particle according to F drag = 6πηa ν, where η is the dynamic viscosity of the medium, a is the radius of the particle and is Faxen's correction factor. We perform the experiments at a sufficient distance from the coverslip (8-10 µm) such that ≈ 1 and therefore does not affect (8) appreciably [23]. Following O'Neil and Padgett [28], a fully calibrated three-axis piezo-electric stage is used to vibrate the trapped particle with progressively increasing speed in the lateral and axial directions until it is ejected from the trap. These velocities are used to find the applied viscous drag force from which Q axial and Q lateral are calculated. This entire process is repeated 15 times for each input beam. To ensure accurate efficiency measurements, the power transmitted through the objective is measured using a sample cell filled with distilled water. We find that the power at the sample is 0.9 mW, giving 30% transmission of the 3 mW incident beam.

Results and discussion
Figure 4(a) shows the experimental trapping efficiency data taken over 15 runs, and the results are summarized in figure 4(b).
For the axial case, the radial beam has the largest trapping efficiency, which is 1.27× that of the Gaussian input beam. Moreover, we note that this result is consistent with what has been reported in the literature [20]. The two π-phase beams exhibit nearly the same trapping efficiency (∼0.120) and perform comparably with the Gaussian. Overall, the azimuthal beam is least efficient, giving a Q axial 0.38× that of the radial. With the lateral case, the radial beam efficiency is 0.34× that of the Gaussian and is the largest for all vector beams. The π-phase beams again display congruous behavior, with efficiencies of about 0.25× that of the Gaussian. Lastly, the azimuthal beam performs quite poorly with respect to the other beams, showing a lateral trapping efficiency of 0.13× that of the Gaussian.
The axial performance of the various beams can be explained by examining the relative values of their z polarizations ( figure 1, top row). The experimentally obtained axial trapping efficiencies follow exactly the order of simulated axial intensity magnitude in each beam. Most notably, the radial beam, which has the largest z polarization, produces a Q axial that is 2.6× that of the azimuthal beam, which exhibits an identically zero z polarization. The two π -phase beams produce identical amounts of z polarization, at 0.20× that of the radial, and correspondingly yield similar axial trapping efficiencies. Furthermore, these two beams give slightly reduced performance compared to the Gaussian, which produces a larger axial intensity, at 0.35× that of the radial.
The role of spot size on the axial performance of the various beams appears relatively small compared to the role of z polarization. From figure 2(b) it can be seen that the π-radial (π-azimuthal) and Gaussian beams have spot sizes of ∼900 and 450 nm, respectively. However, the π -phase beams perform comparably to the Gaussian in the axial direction. This fact suggests that z polarization is the dominant factor in providing a large Q axial . This is further evidenced by examining the radial beam, which produces 2.8× the z polarization and 1.27× the Q axial for the Gaussian beam, while having a comparable focal spot size.
Trapping among vector beams in the lateral direction is consistent with what is expected given the focal spot size. Namely, the radial beam has a smaller spot size than all the other vector beams and yields a 1.45×, 1.29× and 2.6× increase in Q lateral compared to the π -radial, π-azimuthal and azimuthal beams, respectively. Further, while all vector beams have similar maximum intensity gradient values, the modulation of the intensity gradient in the focal plane for the π -radial, π -azimuthal and azimuthal beams likely leads to their lower Q lateral values in comparison with the radial beam. The largest Q lateral is observed for the Gaussian input beam (0.027), which is expected since it produces the largest intensity gradient.

Conclusion
Our results confirm that the axial trapping efficiency can be optimized by the amount of z polarization produced in the focal plane. Further, we find that even in a low-power trapping environment, a radial vector beam is advantageous over a Gaussian beam for axial trapping, and the π -phase vector beams produce axial trapping efficiencies comparable to that of the Gaussian. Additionally, this study demonstrates that the relative phase between the eigenmodes comprising the vector beams is an important parameter for tuning the trapping force applied by the vector beams. Currently, we are investigating the trapping efficiency for vector beams with relative phases other than π , which may result in an input beam that optimizes both Q lateral and Q axial .