1. Introduction

Optical tweezers have attracted extensive academic interests since the first demonstration [1, 2]. Experimental and numerical researches have been implemented to investigate the optical forces of a variety of laser beams on dielectric particles [3-11]. In these works, Rayleigh scattering model and ray optics model were used to cope with Rayleigh particles (a << λ where a is the particle radius) and particles with radius a >> λ, respectively. For particle size in the intermediate regime between the above two cases, finite-difference time-domain method (FDTD) and T-matrix method have been introduced to obtain quantitative results.

However, the trapping situation becomes complicated when a metal particle is used. It is known that optical trapping of metal particles with size beyond the Rayleigh regime is more difficult, compared to their dielectric counterpart, due to significantly strong scattering and absorption in metal particle. If the size of metal particles is comparable to or larger than the trapping wavelength, trapping in most cases can be achieved only in two dimensions, and three-dimensional trapping is possible only by using some elaborated experimental arrangements and tricks [12-16]. Due to complexity in electrodynamic interactions between metal particle and illumination beam, there have been relatively less reports on theoretical studies on optical trapping of metal particles. Moreover, most evaluations for the trapping forces on metal particles are still restricted to cases of either Rayleigh particles or particles with radius a>>λ[17-22]. There has been no detailed characterization of optical force for moderate-sized metal particles whose radii are comparable to wavelength. The aim of this paper is to characterize the optical force on moderate-sized metal particles through numerical approach. We developed an improved finite-difference time-domain (FDTD) method to compute electromagnetic field distribution around the metal particle, and to evaluate the trapping capability of optical tweezers for metal particle. Some unique properties of metal particle within optical traps of various illuminations have been demonstrated.

2. FDTD approach

FDTD modeling is a powerful tool for solving electrodynamic problems, and can be divided into two broad categories: total field method [23] and scattered field method [24, 25]. For calculating optical forces of tightly focused beams on microscopic particles, the scattered field approach is superior to the total field modeling. This is owing to the fact that the computation space in the scattered field regime can be limited to only the region encompassing the particle, and thus computational burden can be greatly reduced, which is crucial for us to run our program on a personal computer, otherwise supercomputer may be needed if one adopts the total field approach. In addition, the scattered field scheme allows the incident field to be calculated separately, which is particularly suitable for optical trapping system in which the calculation of focal field distribution can be readily fulfilled before implementation of FDTD procedure. In this paper, we compute the incident fields by using the Richards-Wolf vectorial diffraction theory to calculate the tightly focused beams near the focal region of an aplanatic lens [26], and then implement the scattered field FDTD to evaluate optical force of vector beams of interest on metal particles. The biggest advantage of combining Richards-Wolf theory with the scattered field FDTD formalism is of enabling one to efficiently compute the field of arbitrary particle illuminated with arbitrary beam. Another problem associated with metal object is that a complex-valued dielectric function and material loss should be taken into account in FDTD approach; that is, the time dependence of a material polarization should be considered. To accomplish this, an improved Drude-Lorentz model for gold dispersion proposed by Vial et al [27] is used in our numerical implementation, and furthermore, we incorporate time-domain additional differential equations (ADE) into FDTD models because that the ADE method has the second-order accuracy and allows us to formulate a memory-efficient set of updating equations. Finally, Maxwell’s stress tensor is introduced to calculate the forces on the particles using the total field which is simply a sum of the incident field calculated from the Richards-Wolf theory and the scattered field computed from the FDTD algorithm. The time-averaged force acting on a particle is given by the integral of stress tensor over the surface enclosing the particle.

3. Numerical results

In this paper we will focus on the investigation into optical forces of tightly focused vector beams illuminating moderate-sized gold particles whose radii are on the order of several hundred nanometers, although the proposed approach is also applicable to other particles. Simulation parameters used in our numerical modeling are specified here: the refractive index of the ambient medium encompassing gold particles is 1.33; a laser beam with the total power of 100 mW is incident on an objective lens with numerical aperture of 1.26. We assume a uniform illumination beam in our simulation for simplicity, and the proposed scheme is also suited to other practical beams such as a Laguerre-Gauss beam. The accuracy of the FDTD method depends mostly on the dimension of the cell within the grid space, so we choose 2.5nm as the cell size after taking into account a compromise between accuracy and computation expense.

3.1 Optical forces of incident light beams with different polarization configurations

Firstly, we numerically explore the trapping efficiency of incident light beam with different polarization configurations, including circular, linear, azimuthal and radial polarizations. Herein we restrict our approach to the situation that the center of the gold particles are localized within the focal plane, otherwise the particle would suffer different optical forces, which will be discussed later in Section 3.4. The wavelength of light is assumed to be 1064nm. The total optical field is obtained by conducting FDTD simulation for tightly focused beams illuminating a gold particle of 100 nm radius, and then the optical force is calculated.

Figure 1 shows curve of transverse component of force exerted on the 100 nm radius gold particle located in the focal plane. It is known that a linearly polarized beam, tightly focused by the objective lens, will produce a noncircular symmetric field distribution in the focal plane. For the x-directional, linearly polarized beam, we chose two typical directions (x and y) in the focal plane to evaluate the transverse trapping efficiency. We place the center of the gold particle in different positions of x and y axes, respectively, and calculate the radial components of force (denoted by F_x and F_y in Fig. 1, respectively). For other beams (with circular-symmetric polarization), we only need to calculate transverse forces on the particle located in different transverse positions denoted by the radial coordinate r of the focal plane.

From Fig. 1 it is clearly seen that all beams, except the azimuthally polarized illumination, can provide transverse restoring forces pointing towards the focus. The azimuthally polarized illumination pushes the particle away from the beam axis due to its hollow distribution. The trapping force exerted on the gold particle behaves like that of dielectric particle, in which the gradient force plays a dominant role. The x-polarized beam produces the steepest potential well for the gold particle located in y axis, and the widest well for the particle in x axis. That means that the x-polarized beam offers bigger radial trap stiffness for the gold particle in y axis than in x axis. We ascribe this difference to polarization-dependent focusing properties. It is known that a tightly focused, x-polarized light produces a focal spot with narrower width in y direction than in x direction [26]. Thus it is reasonable that the gold particle is subjected to a stronger gradient force along y axis than along x axis. The trap stiffness of circular polarization is between the x- and y-polarized beams. This also means that, for a gold particle confined by optical tweezers, average behavior of circular and linear polarizations in transverse trapping will be nearly the same. Assuming the potential, experienced by the trapped particle near the focus (equilibrium position), to be approximately harmonic, we can estimate the radial trap stiffness of circular polarization to be 15.7 pN/μm. The plot also shows that the transverse stiffness of the radial polarization is smaller than that of the circular polarization.

 

Fig. 1. Transverse component of force on the 100 nm radius gold particle located at the focal plane with x and y coordinates, illuminated with 1064nm light of circular, linear(x-), azimuthal and radial polarization with the total power of 100 mW.

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We then evaluate the axial trapping efficiencies of circular, azimuthal and radial polarizations. Figure 2 presents axial components of force exerted on the 100 nm radius gold particle moving along the optical axis. The linear polarization is not included here because its result turns out to be almost the same as the circular polarization. Besides, the simulation results of transverse and axial forces indicate that the linear polarization has much the same three-dimensional trap stiffness as the circular polarization on average. The solid- line-symbol curves in Fig. 2 represent the axial forces produced by the three kinds of polarization beams. From the three force distributions, we know that axial trapping can not be achieved at focus for the 100nm gold particle. It is noted that the force curve of radial polarization has an axial trapping position (the point of the curve intersecting with z-axis in Fig. 2) near z=0.7 μm, which suggests a possibility of three-dimensional trap for such a particle. For more elaborate analysis, the force can be considered as contributions of two respects, i.e., gradient force and repulsive force resulting from absorption and scattering of metal particle. Taking into account the nature of the two forces, we divide the force curve into even-symmetric and odd-symmetric components, shown by dashed and dotted lines in Fig. 2, respectively. It is reasoned that the gradient force, pointing towards the maximal intensity, is responsible for the odd-symmetric component, and the repulsive force, which propels particles moving in the direction of beam propagation, contributes to the even-symmetric component. The axial force of azimuthally polarized light is rather small compared to the others, and contributes mainly to push the particle away from the dark focal point generated from tightly focused, azimuthally polarized beam. For both circular and radial polarizations, the repulsive component remains dominant. However, the radially polarized beam can produce equal restoring force but much smaller repulsive force, compared with the circular polarization. This implies radial polarization is more suitable for axial trapping such a gold particle. The reason is ascribed to the fact that the strong axial component of the tightly focused radially polarized beam does not contribute to the Poynting vector along the optical axis and thus gives rise to a weaker axial scattering and absorption [20]. Since axial trapping is normally more difficult than transverse trapping in optical tweezers, this result indicates that the radial polarization can greatly benefit optical trapping of moderate-sized metal particles, even though, as shown in Fig. 1, the transverse stiffness of the radial polarization is smaller than the circular polarization.

 

Fig. 2. Axial components of force exerted on the 100nm radius gold particle located at the optical axis with 1064 nm light illumination of circular (cir), radial (rad) and azimuthal (azi) polarizations with the total power of 100 mW. The solid- line-symbol curve represents the total force, which is divided into even-symmetric and odd-symmetric components, shown by dashed and dotted lines, respectively.

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We have also calculated the axial forces produced by circularly and radially polarized illuminations when particle radius goes to regime below 100 nm, and confirmed that the above statement holds true for a 50 nm radius gold particle. As an example, the result of 50 nm gold particle is plotted in Fig. 3. Comparisons between Figs .2 and 3 show that, although the total axial force on 50 nm particle is about 0.15 of that on the 100 nm particle, the ratio between the restoring force and repulsive force increases from 0.12 for 100 nm radius to 0.21 for 50 nm radius, which implies that the axial trapping may be possible if the particle radius continues to decrease. The reason behind this inference is that the smaller the particles, the smaller the cross sectional area and, hence, the smaller the scattering force. This observation is also consistent with the experimental results of optical trapping in the Rayleigh regime [18].

However, Fig. 3 also shows that the axial trapping of 50 nm gold particle is not yet possible, which seems inconsistent with experimental trapping of gold particles with radii up to 127 nm, reported by Hansen et al [16]. In their paper, Hansen et al emphasized a main experimental trick for three-dimensional trapping of gold particle by stating that it was crucial that the tails of the Gaussian beam profile were not cut off by any optics. We guess that our negative result about axial trapping for 50 nm gold particle might result from the over-simplified illumination condition that assumes a uniform illumination beam incident on the objective lens, and illumination arrangement could wield significant influence on the axial distribution of light field and, thus, dominate axial trapping behavior. The further investigation into influence of illumination arrangement on trapping is still under study.

In order to verify the validity of our calculation, Fig. 3 also plots the curve of transverse force exerted on the 50 nm radius gold particle located in the focal plane, shown by the black solid line. The transverse trapping stiffness in this case is estimated to be 4.0 pN/μm, which is about 1/4 of 100 nm radius gold particle. This result is in agreement with the trapping model of gold particle illustrated in [16, 18]; the gradient force is proportional to a ² (for particle with radius, a, larger than 50 nm) or a ³ (for particle with radius smaller than 50 nm). Furthermore, the trapping stiffness, measured by Svoboda and Block [18], of 18 nm radius gold particle at 1047 nm wavelength was 0.5 pN/μm, in contrast to our calculated value, 4.0 pN/μm, for 50 nm radius gold particle at 1064 nm wavelength. Hence, our calculation shows that the transverse trapping stiffness of 50 nm radius gold particle is nearly octuple as much as that of 18 nm radius gold particle. Bearing in mind that the square of 50/18 is 7.7, our calculated value is thought to comply with the above trapping model in the case of transverse trapping.

 

Fig. 3. Force exerted on the 50 nm radius gold particle under 1064 nm illumination of circular polarization with the total power of 100 mW. The blue solid-line-symbol curve represents the axial force when particle moving along optical axis. The axial force is divided into repulsive and attractive components, shown by green dashed and red dotted lines, respectively. The black solid line shows transverse force exerted on the 50 nm gold particle when it moving within the focal plane.

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3.2 Influence of particle size on transverse trapping efficiency.

Our next concern is the influence of particle size on transverse trapping efficiency. We present in Fig. 4(a) the simulation result of transverse trapping forces on 200nm, 250nm and 300nm gold particles under circular polarization illumination with 1064 nm wavelength, and in Fig. 4(b) the simulation result of transverse trapping forces on 50nm, 75nm, and 100nm gold particles from the same polarization with 532nm wavelength, respectively. It can be seen from Figs. 4(a) and 4(b) that for both beams there exists a critical radius of particle beyond which no particle could be trapped. The transverse force is seen to vary sign (attractive to repulsive) around the critical radius. We believe that this phenomenon arises from the competition and balance between the gradient and the scattering forces; the gradient force overwhelms the scattering one for smaller particles, and vice versa for larger particles. The critical radius for 1064 nm wavelength is above 200nm, being much larger than the 532 nm’s critical radius that is nearly 75nm. This means longer wavelength beam is able to trap larger metal particle. An intuitive explanation on this behavior is that, longer the wavelength, wider the focal spot, wider the potential well and, thus wider the trapping scope, being helpful for trapping larger metal particles. It is of interesting that the force curve of 75 nm in Fig. 4(a) has a plateau region in vicinity of focus, wherein the gradient force and the scattering one counteract, thus giving rise to zero transverse optical force. Fig. 4(c) shows the transverse forces produced by 1064 nm radially polarized illumination on 300 nm, 400 nm and 450 nm gold particles. A critical radius of the radially polarized illumination is thought to exist between the range from 400 nm to 450 nm. It is interesting that the radially polarized illumination can trap a larger gold particle than the circularly polarized illumination, which implies that vectorial property of polarized light can play a great role in metal particle trapping. The fact that the vector beam can offer added benefit of enhanced trapping has also been demonstrated theoretically elsewhere [3, 20].

 

Fig. 4. Transverse force components: (a) 200 nm, 250 nm and 300 nm radius gold particles illuminated by circular polarization of 1064 nm wavelength; (b) 50 nm, 75 nm, and 100 nm radius gold particles illuminated by circular polarization of 532nm wavelength; (c) 300 nm, 400 nm and 450 nm radius gold particles illuminated by radial polarization of 1064 nm wavelength.

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3.3 wavelength-dependence of the transverse trapping forces.

In the previous section, we have observed difference in transverse forces between 1064 nm and 532 nm wavelengths. This suggests that it would be helpful through further exploring the wavelength-dependence of the transverse forces exerted on gold particle. Now we investigate circularly polarized beams with wavelengths of 532 nm, 633 nm, 780 nm, 1064 nm, which are most frequently used in optical trapping experiments. The 600 nm wavelength is also included in order to figure out a transition phenomenon in trapping property. The transverse forces of a 125 nm radius gold particle are shown in Fig. 5. The figure shows that 633 nm, 780 nm and 1064 nm circularly polarized laser beams can provide a stable trapping for the gold particle at the focal point, but 600 nm and 532 nm laser beams can not trap such a gold particle. This confirms again that longer wavelength leads to bigger critical radius and 125 nm radius of gold particle goes beyond critical radii of 600 nm and 532 nm laser beams. Between 600 nm and 633 nm there exists a transition range in which the status of particle transforms from being trapped into escaped. We attribute the arising of the transition region to the dispersion property of gold [27]. In fact, from dispersion curve (see Figs. 1 and 2 of Ref. 27), the imaginary part of permittivity of gold reaches a minimum between 600 nm and 633 nm, while the real part changes monotonically with wavelength. Below 600 nm, the imaginary part increase rapidly, which is responsible for a relatively large absorption force, tending to destabilize the trap. In addition, it is noted in Fig.4 that the 600 nm wavelength seems able to provide a stable trap for the gold particle along a ring encompassing the beam axis, underlying physical mechanism in which needs further investigation. On the whole, it turns out that, among above mentioned wavelengths, 780nm-wavelength is more suitable for transverse trapping of the 125 nm gold particle.

 

Fig. 5. Wavelength-dependence of the transverse trapping forces exerted on 125nm radius gold particle located at the focal plane, illuminated by circularly polarized light at 532 nm, 600 nm, 633 nm, 780 nm and 1064 nm wavelengths.

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We have also noted that the two-dimensional trapping of gold particles with diameters from 0.5 to 3 μm has been reported by using a linearly polarized laser beam with the wavelength of 515 nm [19]. The experimental result seems inconsistent with our calculation shown in Fg. 5. However, such a two-dimensional trapping could only be possible in planes before the focus, as illustrated in Ref. 19. Hence, it is appealing that trapping capability outside the focal plane should also be studied, which will be provided in detail in the next section. In order to eliminate the possible misleading evoked by Fig. 5, here we present in Fig. 6 a numeric calculation for a 125nm radius gold particle, located at three different planes before the focal plane and illuminated by 532 nm circularly polarized light. Figure 6 shows that the 125nm radius gold particle could be trapped within planes with distance longer than 0.4μm before the focal plane. This calculation confirms that the two-dimensional trapping of gold particles could take place within planes before the focus, even though on-focal-plane trapping is not possible. The explanation to this phenomenon will be given in the next section.

 

Fig. 6. Transverse trapping forces exerted on 125nm radius gold particle, located at three different planes before the focal plane, illuminated by 532 nm circularly polarized light. The incident beam propagates along +z axis and is focused on the focus. Two-dimensional trapping becomes possible for the planes with distance 0.4mm and 0.5 mm before the focal plane.

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3.4 Trapping capability outside the focal plane

Finally, we investigate the trapping capability of focused beam for metal particle outside the focal plane. In all above simulations except Fig. 6, the centers of gold particles are assumed to be located in the focal plane, which yields an observation that the trapping of metal particle beyond the critical radius appears to be impossible. Actually, when one tries to trap a particle with optical tweezers in experiments, the total focal volume of focused beam should be taken advantage of. Hence a three-dimensional picture of optical trapping should be considered. For this purpose we choose a 300 nm radius gold particle to investigate optical forces exerted by the 1064 nm circular polarized beam. We have calculated optical forces within a series of planes before and behind the focus.

 

Fig. 7. Axial position-dependence of the transverse trapping forces exerted on gold particle of 300nm radius, illuminated by 1064nm circularly polarized light. (a) Particle was placed before the focal plane. (b) Particle was placed behind the focal plane.

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The simulation in Section 3.1 has indicated that, if the gold particle is small enough, it can be trapped at the focus, because the gradient force plays a dominant role in trapping. By contrast, Fig. 7(a) shows that if the particle becomes larger, there still exists a possibility of two-dimensional trapping before the focal plane. One intuitive explanation is that the direction of energy-flow carried by tightly focused and off-axial ray before the focal plane points toward the beam axis. When the particle locates at such off-axial position, scattering and absorbing the illumination light, it will encounter a net repulsive force toward the beam axis. On the contrary, based on the same reasoning, the metal particle locating behind the focal plane will be pushed away from the beam axis. In addition, the energy flow near the waist of focused beam has major axial but less transverse components, and thus fails to trap metal particle.

4. Conclusion

In summary, optical forces of tightly focused vector beams on moderate-sized metal particles are characterized. From numerical simulation we have found the critical radius of particle under transverse trapping in the focal plane, and illustrate the dependence of trapping ability on wavelength of incident light. We show that the trapping efficiency can be improved through adjusting the polarization of the incident beam. Besides, we also demonstrated the trapping possibilities of metal particle outside the focal plane. The simulation results provide new insights into how to implement optical micromanipulation for moderate-sized metal particles. We also wish to point out that exactly the proposed approach can be used with dielectric particles.