## Abstract

We present the theoretical analysis and the numerical modeling of optical levitation and trapping of the stuck particles with a pulsed optical tweezers. In our model, a pulsed laser was used to generate a large gradient force within a short duration that overcame the adhesive interaction between the stuck particles and the surface; and then a low power continuous-wave (cw) laser was used to capture the levitated particle. We describe the gradient force generated by the pulsed optical tweezers and model the binding interaction between the stuck beads and glass surface by the dominative van der Waals force with a randomly distributed binding strength. We numerically calculate the single pulse levitation efficiency for polystyrene beads as the function of the pulse energy, the axial displacement from the surface to the pulsed laser focus and the pulse duration. The result of our numerical modeling is qualitatively consistent with the experimental result.

© 2005 Optical Society of America

## 1. Introduction

Optical tweezers is a single-beam gradient force optical trap, formed by tightly focusing a laser beam with an objective lens of high numerical aperture (NA). It has become a powerful tool for trapping and manipulation of dielectric and biological micron-sized particles typically using continuous wave (cw) laser[1–4]. It has been routinely applied to manipulate living cells, bacteria, viruses, chromosomes and other organelles [5–6], and recently been applied to the study of molecular motors at the single-molecule level [4,7] the physics of colloid systems [8, 9], the mechanical properties of polymers [10–11], and the control of optically trapped structures [12–13]. The combination of near-infrared Raman spectroscopy with optical tweezers allows characterizing optically trapped living cells and other particles [14–17]. Agate and co-workers have demonstrated that femtosecond optical tweezers are just as effective as cw optical tweezers [18]. The trapping force generated by the cw optical tweezers is typically in the order of 10^{-12} N [4]. This weak force is efficient to confine micro-particles suspended in liquids, but not sufficient to levitate the particles that are stuck on the glass surface, where they have to overcome the binding force. Recently Ambardekar and co-worker have demonstrated that both the stuck dielectric and biological micron-sized particles can be levitated and manipulated with a pulsed optical tweezers [19]. In their experiment, an infrared pulse laser at 1.06 µm was used to generate a large gradient force (up to 10^{-9} N) within a short duration (~45 µs) that overcame the adhesive interaction between the stuck particle and the surface; and then a low-power cw diode laser at 785 nm was used for trapping and manipulating the levitated particle.

Here, we present a theoretical analysis and the numerical modeling of optical levitation and trapping of stuck particles with a pulsed optical tweezers. We describe the radiation force generated by the pulsed optical tweezers as a pulsed gradient force and describe the binding interaction between the stuck beads and glass surface as the dominative van der Waals force with a randomly distributed binding strength. The equation of motion for the bead will be used to describe the trajectory of the bead’s position as the stuck bead is detached from the surface and moves to the trap center. We will calculate the single pulse levitation efficiency for polystyrene beads as the functions of the pulse energy, the axial displacement from the surface to the pulsed laser focus and the pulse duration by solving the equation of motion for the polystyrene bead.

## 2. Theory

#### 2.1 Pulsed optical tweezers

The pulsed optical tweezers is composed of two parts: the cw trapping system and the pulsed levitation system. The focus of the pulsed laser beam is at the same position as the focus of the cw trapping beam. The configuration is shown in Fig. 1. The polystyrene bead radius is *a*, and the distance from the bottom of bead to the surface is *h*, which is very small when the bead is “stuck” on the surface. In order to detach the stuck bead from the surface with the pulsed tweezers, the focus of the both pulsed and cw beam is initially adjusted to have an axial displacement *z*_{0}
from the surface of the bottom glass plate. Then, a pulse is fired to generate large gradient force acting on the stuck bead within the pulse duration. This pulsed gradient force will break the binding interaction between the bead and the surface so that the stuck bead is levitated and moves into the focus of the cw beam.

Figure 2 shows the image of a 2-µm polystyrene bead that sticks on the glass surface before the application of a pulse (the image was defocused with an axial distance z) and the image of the levitated bead that was detached with the application of a pulse and moved to the focus of the cw beam [19].

#### 2.2 Gradient forces

In order to describe the equation of motion for the bead, we begin with modeling the radiation forces generated by the cw and the pulsed beams that were applied to the bead.

Assume that the beam intensity of the focused laser beams can be described with a Guassian expression.

$$+\frac{2U}{\tau \sqrt{\pi}}\frac{{\omega}_{0}^{2}}{\omega {\left(z\right)}^{2}}\mathrm{exp}(-{(t\u2044\tau )}^{2})\mathrm{exp}(-2\left({x}^{2}+{y}^{2}\right)\u2044\omega {\left(z\right)}^{2}),$$

where the first term describes the cw beam with an intensity I_{0} and the second term describes the pulse beam with the single pulse energy *U* per unit cross-section area, the pulse duration *τ*. Here we assume that the pulse beam is completely overlapped with the cw beam and they have the same spatial intensity distribution. *ω(z)* is the radius of the beam waist at point z along the light propagation and it can be expressed as

where ω_{0} is the radius of the beam waist at the point *z*=0 and z_{s}=π${\mathrm{\omega}}_{0}^{2}$/λ_{0} is a distance parameter. The radiation force acting on the bead includes the scattering force *F*_{scatt}
and gradient force *F*_{grad}
. In the levitation experiment with pulsed optical tweezers, the stuck bead is not at the focus of the pulsed beam but with an axial displacement. Thus, the scattering force can be neglected comparing to the gradient force. Ashkin has calculated the forces of a single-beam gradient laser trap on a dielectric sphere in the ray optical regime [3]. Here, we use an approximate expression for the gradient force. It is known that the gradient force arises from intensity gradient of the inhomogeneous field, *F*_{grad}
∝∇*I*(*x,y,z*) [4]. Here, we approximately express the gradient force acting on the bead that is located on the beam axis (with x=0 and y=0), produced by the cw beam as

where *k*_{z}
is the spring constant and *z* the position deviation of the bead center from the beam focus (*z=h+a-z*
_{0}). The express in Eq. (3a) is consistent with the harmonic approximation of an optical trap, in which the restoring force acting on the bead is described as -*k*_{z}*z* for z≪z_{s} [4]. Eq. (3a) is a good approximation for z<0.5z_{s}, but it can not correctly represent the actual gradient force for z>0.5z_{s} where the force decreases sharply, so we must multiply a correction term

Similarly, we approximately represent the gradient force produced by the pulsed beam as

where *E* is a parameter in proportion to the single pulse energy *U*.

#### 2.3 The equation of motion

As a pulse is fired to the stuck bead, the bead will be detached under the action of the pulse gradient force and then moves to the trap center under the action of the cw gradient force. The equation of motion describing this levitation process of the bead can be represented as

where m is the mass of the bead (m=(4/3)*πa*
^{3}
*ρ* and *ρ* the density of bead), F_{S} the viscose force, FV the van der Waals force. Here, we have neglected the gravity force of the bead and the random force due to Brownian motion that are much smaller than the gradient forces. In addition, we have modeled the adhesive interaction between the stuck bead and the surface by the van der Waals force because it dominates at short distance comparing to the electrostatic force [20].

The van der Waals force, F_{V}, can be approximately expressed as [20]

where *f*(*p*)=(1+3.54*p*)/(1+1.77 *p*) for p<1 and *f*(*p*)=0.98/*p*-0.434/*p*
^{2}+0.067/*p*
^{3} for p>1 with *p*=2*πh/λ*_{L}
, *λ*_{L}
is the London retardation wavelength, usually of the order of 100nm, *A* is the Hamaker constant, typically in the order of 10^{-19}~10^{-21}J [20].

The viscose force, F_{S}, exerted on the bead due to the viscosity of the suspending medium is given by

where D is the damping coefficient (D=6*π*a*ηλ*), *η* the viscosity of the surrounding medium, and *λ*a correction term that depends on the proximity of the bead to a planar boundary surface such as the cover slip or microscope slide. For the motion near a planar boundary, this position-dependent term is represented by [21]

The equation of motion Eq. 5 can be solved numerically on the initial value condition when t=0, h=10^{-9}m, and *ḣ*=0 for the bead’s position at different time after a laser pulse is introduced to a target stuck bead.

## 3. Results and discussion

#### 3.1 Levitation trajectory

As a pulse is fired at t=0, we can obtain the position h (or z) of the bead as the function of time (the levitation trajectory) by solving the equation of motion Eq. (5) numerically. As an example, we selected the parameters for the polystyrene bead as *a*=2*µm*, *ρ*=1.0495×10^{3}
*kg/m*
^{3}, *λ*_{L}
=100nm, the viscosity of water as η=10^{-3}Ns/m^{2}, the parameters for the cw laser as k_{z}=10^{-5} N/m, z_{s}=8µm, the parameters for the pulsed laser as τ=45µs, z_{0}=6µm, A=1×10^{-19} J. We found that for given values of A=1×10-19 J, cw trap force constant k_{z}, sphere size a, and initial displacement z_{0}, if the value of E exceeds a threshold value E_{S} (such as 7×10^{-7}Ns/m), the bead’s position can jump from the sticking position (h=10^{-9}m) to the trap position (h=4µm in Fig. 3). However, if the value of E is less than the threshold value E_{S}, the stuck bead cannot be levitated. The threshold value ES can be calculated numerically. This behavior is consistent with the experimental observation [19] Figure 3 shows the distance h of the bead from the glass surface versus the time as a single pulse is illuminated on the bead with the different pulse energy: (a) *E*=1.1×10^{-5}
*Ns/m*; (b) *E*=7×10^{-7}
*Ns/m*; and (c) *E*=0.

#### 3.2 The levitation efficiency versus the pulse energy

The success in levitation relies on the magnitude of the pulsed gradient force and the binding condition between the stuck bead and the surface. For the given bead and aqueous solution, the pulsed gradient force depends on the pulse energy, the pulse duration and the axial displacement *z*_{0}
. Experimentally, we found that even with the same pulsed gradient force (so as the same pulse energy, duration, and axial displacement), some beads can be successfully levitated, but some beads in the same liquid sample cannot be levitated. We attributed this phenomenon to the effect of the variation in binding condition between the individual stuck bead and the surface. This leads to the levitation efficiency at a given pulse energy, the pulse duration, or the axial displacement *z*_{0}
.

We modeled the change in the adhesive interaction between the stuck beads and glass surface among individual beads by considering the Hamaker constant *A* as a randomly distributed parameter, satisfying a normal distribution, with a probability density function of

where *ζ* and *σ*
^{2} are the mean value and the variance of *A*, respectively. The cumulative distribution function of the normal distribution is

where $\Phi \left(u\right)={\int}_{-\infty}^{u}\frac{1}{\sqrt{2\pi}}{e}^{-{t}^{2}\u20442}\mathit{dt}.$.

In the numerical calculation (such as in Fig. 3), we found that for given values of pulse energy E, cw trap force constant k_{z}, sphere size a, and initial displacement z_{0}, if the value of A is less than a critical value A_{M}, the bead’s position can jump from the sticking position (h=10^{-9}m) to the trap position (h=4µm in Fig. 3). However, if the value of A exceeds the critical value A_{M}, the stuck bead cannot be levitated. The critical value A_{M} can be calculated numerically. Therefore, the probability that a stuck bead can be levitated with the given conditions in E, k_{z}, a, and z_{0}, can be calculated by P{A_{M}}, where A_{M} is the maximum value in which the bead can be levitated. Hence, the levitation efficiency is equal to P{A_{M}}.

Figure 4 shows the theoretical levitation efficiency versus the *E* (that is proportion to single pulse energy) for a fixed z_{0}=6*µm* and *τ*=45*µs*. In this calculation, the Hamaker constant *A* was selected with a normal distribution with ξ=1×10^{-19}
*J, σ*=0.5×10^{-19}
*J*. The line curve is the Boltzmann fit of the computational data. One can see that the levitation efficiency increases with the increase in *E* and can reach up to 94.5% at *E*=1.1×10^{-6}
*Ns/m*. As a comparison, shown in the right of Fig. 4, the experimental measurement in the levitation efficiency is found to be qualitatively consistent with the numerical calculation.

#### 3.3 The levitation efficiency versus the axial displacement z0 and pulse duration

We also calculate the levitation efficiency versus the axial displacement z_{0} and pulse duration τ. Figure 5 shows the dependence of the levitation efficiency on the displacement z_{0} with the fixed *E*=1.1×10^{-6}
*Ns/m* and *τ*=45*µs*. The optimum efficiency was obtained at z_{0}~6*µm*.

Figure 6 shows the dependence of the levitation efficiency on the pulse duration *τ* with the fixed *E*=1.1×10^{-6}
*Ns/m* and z_{0}=6*µm*. The levitation efficiency decreases with the increase in the pulse duration *τ*. The result of our numerical modeling in Figs. 4–5 is qualitatively consistent with the experimental result [19].

## 4. Conclusion

In summary, we have numerically modeled the optical levitation and trapping of the stuck particles with a pulsed optical tweezers. We described both the pulsed gradient force and the binding interaction force between the stuck bead and the glass surface. We numerically calculated the single pulse levitation efficiency for polystyrene bead as the functions of the pulse energy, the axial displacement from the surface to the pulsed laser focus and the pulse duration. The result of our numerical modeling is qualitatively consistent with the experimental result.

## Acknowledgments

This work is supported by grant No. 022261006 and 036105016 from Shanghai Optical-Tech Special Project.

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