## Abstract

Metallic particles are generally considered difficult to trap due to strong scattering and absorption forces. In this paper, numerical studies show that optical tweezers using radial polarization can stably trap metallic particles in 3-dimension. The extremely strong axial component of a highly focused radially polarized beam provides a large gradient force. Meanwhile, this strong axial field component does not contribute to the Poynting vector along the optical axis. Consequently, it does not create axial scattering/absorption forces. Owing to the spatial separation of the gradient force and scattering/absorption forces, a stable 3-D optical trap for metallic particles can be formed.

©2004 Optical Society of America

## 1. Introduction

It is well-known that the linear momentum of light wave can exert radiation pressure on microscopic physical objects and perform mechanical manipulations on them. Since Ashkin demonstrated the first practical laser traps and showed the use of radiation pressure to capture and manipulate micrometer sized particles[1], optical traps and tweezers have become powerful tools for the trapping and manipulations of atoms, molecules, nano particles, living biological cells and organelles within cells[2].

Two types of radiation forces were identified in the optical tweezers: gradient force and scattering/absorption forces. The gradient force is proportional to the gradient of the square of the electric field (energy density) and is responsible to pull the particles towards the center of the focus. The scattering/absorption forces are due to the net momentum transfer caused by scattering/absorption of photons from the particles. These forces are proportional to the Poynting vector (power density) of the field. If the particle is located at the center of the focus, the scattering/absorption forces tend to push the particles out of the focus and destabilize the optical trap. For dielectric particles, the gradient force can easily surpass the scattering force. Thus, it is relatively easier to trap dielectric particles in 3-dimension (3-D). However, for metallic particle, stable 3-D trapping is generally considered difficult due to strong scattering and absorption. Although trapping of both Rayleigh and Mie metallic particles has been reported previously [3–5], the formation of stable trap have been largely limited to either extremely small particles or need to be optimized at a longer wavelength. As wavelength decreases, the scattering/absorption forces increases much faster than the gradient force, which could potentially destabilize the trap.

Recently, a new method of shaping the focus using highly focused cylindrical vector beams was proposed and its application to optical tweezers was suggested [6, 7]. It has been shown that dielectric Rayleigh particles with different refractive index can be stably trapped using either a strongly focused radial polarization or a strongly focused azimuthal polarization. In this paper, we further show that metallic Rayleigh particles can also be stably trapped in 3-D by a strongly focused radial polarization. In section 2, we illustrate the geometry that is used in our optical trapping calculations and briefly review the focal field calculation for radially polarized beam. We then present the numerical simulations of the radiation forces exerted on metallic Rayleigh particles and show that 3-D stable optical trap can be formed due to a lacking of power density along the optical axis for highly focused radially polarized beam. A short discussion is then given in section 4.

## 2. Focal field calculation of highly focused radial polarization

There is an increasing interest in laser beams with radial polarization. Particular interest has been given to the high numerical aperture (NA) focusing properties of these beams and their applications as high-resolution probes [8–11]. Due to the polarization symmetry, the electric field at the focus of a radially polarized beam has an extremely strong axial component, and the transverse size of the axial component is much smaller than that of the transversal component. This property may find applications in high-resolution microscopy, microlithography, metrology and nonlinear optics, etc. In this paper, we show that this unique focusing property allows stable 3-D trapping of metallic nano-particles.

#### 2.1 Geometry and focal field calculation for highly focused radial polarization

The field distribution near the focus of highly focused polarized beams is analyzed with the Richards-Wolf vectorial diffraction method [12, 13]. Details of focal field calculation for radial, azimuthal and generalized cylindrical polarization using Richards-Wolf method can be found in [6, 10]. The geometry of the problem is shown in Fig. 1. The illumination is a radially polarized beam with a planar wavefront over the pupil. An aplanatic lens produces a converging spherical wave towards the focus of the lens. The particle to be trapped is assumed to be immersed in an ambience with refractive index of n1. For simplicity, the refractive index on the object side is assumed to be the same as the ambience. The focal field can be written as

where *e⃗r, e⃗z* are the unit vectors along the radial and axial directions. The amplitudes of two orthogonal components E_{r}, E_{z} can be expressed as

where θ_{max} is the maximal angle given by the NA of the objective lens, P(θ) is the pupil apodization function, k_{1} is the wave number in the ambient medium, J_{n}(x) is the Bessel function of the first kind with order n, and A=n_{1}πf/λ with f being the focal length. For many applications, such as high resolution imaging, it is sufficient to simply calculate the relative strength of each field components. However, quantitative estimation of the radiation forces requires the calculation of absolute field strength with given beam parameters. In the calculations throughout this paper, we choose λ=1.047µm, n_{1}=1.33 and a simple annulus pupil apodization function

where P_{0} is a constant amplitude factor and NA is the lens numerical aperture determined by the outer radius of the annulus and n_{1}. The NA is chosen to be 0.95n_{1} in this study. NA_{1} corresponds to the inner radius of the annulus which is variable. The laser beam power is assumed to be 100 mW. As NA_{1} is varied, the amplitude P_{0} is adjusted accordingly to maintain the power level. As one example, field distribution for NA_{1}=0.6 is shown in Fig. 2.

#### 2.2 Properties of highly focused radial polarization

From Fig. 2, it can be seen that the overall distribution of field strength is dominated by the axial component, giving a more compact focus. Thus, focused radial polarization provides stronger gradient force to pull the metallic particles towards the center of the focus. Another important feature of such a focal field is that the strong axial component is a non-propagating field and does not contribute to the energy flow along the propagation direction. Consequently, the time averaged Poynting vector <*S*>z=Re{(*E⃗*×*H⃗**)z}/2=Re{*E*_{r}*H**_{φ}}/2 along the optical axis has a null at center. To demonstrate this, the corresponding time averaged Poynting vector is calculated and its axial component is shown in Fig. 3. The magnetic field is calculated in a similar way to that of the electric field calculation; assuming the magnetic field at the pupil plane is aligned along the azimuthal direction. From Fig. 3, it is clear that axial Poynting vector near optical axis is substantially zero. As we show in the next section, this has important meaning for metallic particle trapping in optical tweezers.

## 3. Optical trapping of metallic Rayleigh particles using radial polarization

#### 3.1 Radiation forces on metallic Rayleigh particles

For Rayleigh metallic particles, one can use the dipole approximation to calculate the radiation force [14, 15]. Assuming a spherical metallic Rayleigh particle with radius a (a≪λ), the gradient force on this particle can be expressed as *F⃗*
_{grad}=Re(*α*)*ε*
_{0}∇|*E⃗*|^{2}/4, where *α* is the polarizability of the metallic particles given by *α*=4*πa*
^{3}
*ε*
_{1}(*$\widehat{\epsilon}$*-*ε*
_{1})/(*$\widehat{\epsilon}$*+2*ε*
_{1}), with *$\widehat{\epsilon}$*,*ε*
_{1} being the dielectric constants of the metallic particle and the ambient, respectively. In this paper, we use gold particle in the simulations (*$\widehat{\epsilon}$*=-54+5.9*i* at 1.047 µm [3]).The axial components of scattering and absorption forces are *F*_{scat,z}
=*n*
_{1}<*S*>_{z}
*C*_{scat}
/*c*, and *F*_{abs,z}
=*n*
_{1}<*S*>_{z}
*C*_{abs}*/c*, where <*S*>_{z} is the time averaged axial component of the Poynting vector, C_{scat} and C_{abs} are the scattering and absorption cross sections. For Rayleigh particles, these cross sections are given by C_{scat}=${k}_{0}^{4}$|*α*|^{2}/6*π* and *C*_{abs}
=*k*
_{1} Im(*α*)/*ε*
_{1}, respectively. It is clear that the forces can be divided into two types. The gradient force F_{grad} is proportional to the gradient of the square of the electric field (energy density). The absorption/scattering forces F_{scat} and F_{abs} are proportional to the Poynting vector (power density).

#### 3.2 Formation of 3D stable trap for metallic Rayleigh particles

To this point, the advantage of using highly focused radial polarization for metallic particle trapping should be clear. The strong confinement of axial component contribute to an enhanced gradient force, while the scattering/absorption forces are zero along the optical axis and remains negligible near the optical axis due to the spatial separation of the energy density and power density. Stable 3-D optical trap (optical tweezers) is thus formed.

Simulation results of radiation forces for NA_{1}=0.6 are shown in Fig. 4. In comparison, calculation results with 100 mW linearly polarized light illumination are also included. It can be seen that annular illumination with radial polarization provides higher gradient force as well as lower scattering/absorption forces. Along the optical axis, the scattering/absorption force is substantially zero, eliminating the major potential cause of trap destabilization.

#### 3.3 Trapping stabilities

To obtain stable 3-D trap, several stability criterion need to be satisfied. To balance the forward scattering/absorption forces and stably trap metallic particles in 3-D, gradient force should provide a sufficient large component opposite to the propagation direction, i.e. *R*=*F*_{grad,z}
/(*F*_{scat,z}
+*F*_{abs,z}
)≥1, where R is called the stability criterion. From the previous sections, it is clear that this can be easily satisfied by the optical trap using radial polarization, since the scattering force around optical axis is negligible. For a more conservative estimation of R, we use *R*=(*F*_{grad,z}
)_{max}/[(*F*_{scat,z}
)_{max}+(*F*_{abs,z}
)_{max}≥1 to estimate the stability. We calculated R for three different NA_{1} values of 0.01, 0.6 and 0.8 and the stability parameters are 24.9, 26.2, and 22.8. In comparison, the corresponding stabilities parameter for 100 mW linearly polarized illumination are also calculated to be 10.5 (NA_{1}=0.01), 7.6 (NA_{1}=0.6) and 4.5 (NA_{1}=0.8), respectively. This clearly demonstrates the advantage of using radial polarization over linear polarization.

In addition, in order to form a stable trap, the potential well generated by the gradient forces must be deep enough to overcome its kinetic energy in Brownian motion. A generally accepted criterion is ${R}_{\mathit{Thermal}}={e}^{-{U}_{max}\u2044{k}_{B}T}\ll 1$, where the potential depth is given by *U*
_{max}=*ε*
_{0}Re(*α*)|*E*${|}_{\text{max}}^{2}$/2 [16]. Assuming a temperature of 300K, *R*_{Thermal}
for the situations considered above are calculated to be 3.34 ×10^{-12} (NA_{1}=0.01), 1.21×10^{-12} (NA_{1}=0.6) and 2.57×10^{-9} (NA_{1}=0.8). The corresponding *R*_{Thermal}
for linear polarization illumination are 3×10^{-19} (NA_{1}=0.01), 2.97×10^{-12} (NA_{1}=0.6) and 1.18×10^{-6} (NA_{1}=0.8). We notice a difference from the thermal stability parameter estimated in [3]. This may be caused by different optical models being used. In Ref. [3], a simple Gaussian beam model was used, while we use uniform illumination with rigorous vectorial model. At extremely low NA_{1}, linear polarization illumination actually gives better thermal stability. For intermediate and high NA1, radial polarization illumination has advantages in terms of thermal stability.

Finally, in some cases, the gradient force needs to be large enough to balance the gravity. Using the density of gold (~19.3 g/cm^{3}) and ignoring the buoyancy force, the gravity of a gold nano-sphere with radius of 18.1 nm is 4.7×10^{-18}
*N*, which is much smaller than the maximal gradient force 7.4×10^{-14}
*N* (NA_{1}=0.01), 5.8×10^{-14}
*N* (NA_{1}=0.6) and 2.52×10^{-14}
*N* (NA_{1}=0.8) in the optical trap using radial polarization.

## 4. Discussions

Although 100mW power is used in the previous calculation, the stabilities calculation indicates that much lower power can be used to form a stable 3-D trap. Its advantages over optical trap with linear polarization allow lower laser power to be used. The lower laser power and the low power density along the optical axis lead to lower absorption and make it less likely to damage the sample. In addition, the trap produced by radial polarization is not limited by wavelength. Because of the skin depth, in conventional trap the gradient force grows at a weaker dependence than a^{3} as particle size increases, eventually settling on a^{2}, while the scattering force grows at a dependence of a^{6} [3]. Thus destabilization caused by scattering force becomes increasingly severe as particle size increases. This is not the case for optical traps using radial polarization. Metallic particles with larger sizes can be stably trapped, as long as the particles can fit in the dark hollow region shown in Fig. 3. From the simulation, we can see that intermediate NA_{1} is preferred to achieve higher gradient forces for trapping and better stability. The axial component is mostly contributed by the higher NA portion of the illumination. For lower NA_{1}, the axial component is less significant and the field distribution is more spread out than linearly polarized incident. Thus, for lower NA_{1}, radial polarization does not show advantage in terms of absolute value of gradient force. However, it still provides the advantage of negligible scattering/absorption forces. For extremely large NA_{1}, high sidelobes lead to weaker radiation forces and lower stability.

The capability to trap metallic particles with shorter wavelength, extremely small sizes at low power has significant meaning for bio-imaging. Metallic nano-particles have important applications in florescence bio-imaging to probe specific targets. Trapping these metallic nano-particles allows various spectroscopic methods to be attached to optical tweezers, opening new possibilities in intracellular imaging at single molecule level.

## 5. Conclusions

In summary, optical tweezers using highly focused radial polarization is proposed to trap metallic Rayleigh particles. Calculations show that there is a separation of the gradient force and the absorption/scattering force due to a dominant non-propagating axial component of the electric field. The extremely strong axial field only contributes to gradient force. The spatial separation of the forces and the dominance of the non-propagating axial field allow one to trap metallic particles in 3-D stably.

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