1. Introduction

Since the first report of single beam optical tweezers in 1986 [1], optical trapping has been routinely applied in a wide-ranging series of experiments from the cooling and trapping of neutral atoms to manipulating biological cells including living cells, bacteria, virus, organelle and macromolecules [2–4]. Optical trapping is now to being employed in a number of applications such as the study of molecular motors at the single-molecule level, the physics of colloids and mesoscopic systems, and the mechanical properties of polymers and biopolymers [5–7]. Optical trapping is based on intensity gradient force exerted on a dielectric particle by a focused laser beam. Recent applications include localized dynamic light scattering [8,9], hydrodynamic cross-correlation between optically trapped particles and femtonewton force spectroscopy [10,11], measurement of dynamic mechanics of single molecules [12], optical deformability [13], and Raman spectroscopy of single optical trapped biological cells [14]. In many applications, latex beads were used as a handler [5] and their dynamic motions in optical traps were measured either in the axial or transverse directions. When a single dielectric particle is trapped at the focus of a Gaussian beam (TEM₀₀ mode) propagating in z direction, the particle undergoes Brownian motion within a confined region due to the random force from the surrounding solvent molecules. The potential well generated by the trapping laser is approximately treated as harmonic [3,8–10], with a restoring force -k_ir_i acting on the particle, where k_i is the spring constant and r_i the position deviation in the transverse direction (with i=x or y) or in the axial direction (with i=z). It was shown that the confinement of Brownian motion in z direction is much weaker than that in the transverse direction (e.g. k_z<k_x, k_y) due to the weaker intensity gradient in axial direction than that in the lateral directions [3]. It is usually considered that the confinement in transverse direction is symmetry (e.g. k_x=k_y) since the laser intensity profile is symmetry in x-y plane for a Gaussian beam. Under this approximation, the particle’s position fluctuations in two lateral directions are decoupled and uncorrelated [8], and, therefore, the center of mass of the particle is randomly distributed in a circular area on the transverse plane.

However, most biological or colloidal particles in a solution environment are not simply dielectric but posses an electrical charge on their surfaces so that they are surrounded by a diffuse ionic double layer [15–17]. When a linearly polarized laser field interacts with the trapped colloidal particle, the ionic molecules in the double layer will redistribute and then affect the dynamic motion of the colloidal particle. Here we are interested in the way the ionic double layer that affects the Brownian motion of the charged colloidal particle in a linearly polarized optical trap. We measured the dynamic motion of a trapped charged colloidal particle captured in a linearly polarized optical trap. We found that the position’s fluctuation of a trapped particle in the parallel direction to the laser polarization is larger that that in the perpendicular direction to the polarization. This suggests that there exists an additional electric force parallel to the laser polarization direction acting on the charged particle beside the known laser forces on the dielectric particle. In an optical trap formed by a circularly polarized beam, this asymmetry in dynamic motion disappears. We will address this problem both experimentally and theoretically.

2. Experimental setup

The experimental setup is shown in Fig. 1. In our experiment, an infrared beam from a continuous-wave Nd:YAG laser (λ=1.06 µm) was expanded with a 5x beam expander and then introduced in an inverted microscope equipped with a high NA objective to form a single-beam optical tweezers. The incident light is linearly polarized with a TEM₀₀ transverse mode and the beam size at the focus is diffraction-limited (with a waist about 1 µm). The polarization direction of the incident beam can be changed with rotating a λ/2 waveplate. The incident laser power can be increased up to 50 mW measured before entering the objective lens. A single polystyrene spherical particle in water was trapped in beam waist area. A CCD video camera was used to confirm the trapping. The dynamic movement of the trapped particle was measured with a quadrant photodiode detector (QD). The backscattered light from the trapped particle was imaged onto the quadrant detector with an optical magnification of ~1500x. The outputs from the quadrant detector that represent the position of the particle in the trap were acquired synchronously at a rate of 50 kHz, allowing to measure the particle’s x and y displacements simultaneously. The x and y directions were determined by the orientation of the quadrant detector while the polarization direction of the beam can be rotated. The quadrant detector was mounted on a precision translation stage so that the particle’s position in the trap can be calibrated by measuring the particle’s offset from the detector center and the optical magnification. Subsequent data processing included subtraction of the offset resulting from the long-term drifts of the experimental apparatus and normalization of the photodiode intensity to the incident laser power [10]. Eventually, the position of the mass center of the trapped particle on the transverse plane can be plotted with each data point of (x, y). The autocorrelation functions for x and y displacements as well as their cross correlation function were calculated. The intensity profile of the incident laser beam was circular symmetry on the transverse plane, which was confirmed by measuring the spot of the laser beam with a CCD camera before and after the objective lens.

 

Fig. 1. Experimental setup. A Gaussian beam from a Nd:YAG laser source is expanded (5 x) and introduced into an inverted microscope with a high NA objective (obj.). The polarization of the laser beam can be changed with a λ/2 waveplate. The backscattered light from the trapped particle is imaged onto a quadrant detector (QD). The inserted picture is the image of the backscattered light from a 0.3-µm polystyrene sphere on the QD surface. BE — beam expander, BS — beam splitter, DM — dichroic mirror, Obj — objective, λ/2 —half wavelength plate.

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The colloidal particles in our study are polystyrene spheres with a diameter of d=0.3 µm, suspended in de-ionized water. When such spheres are dispersed in water, the ionic groups bonded to their surfaces dissociate and give rise to a screened electrostatic interaction [15]. The polystyrene particles in water have a titratable charge (~0.1 e/nm²) on their surfaces and the effective charge is Z*=C(α/λ_B), where λ_B=0.715 nm is the Bjerrum length for a 1:1 electrolyte in water, C is a constant around 10, and α is the radius of the colloidal sphere [16,17]. For a 0.3-µm polystyrene sphere, Z*~1.4×10³ e. This effective charge is negative. Around the negatively charged polystyrene sphere, a positive ion double layer is formed.

3. Experimental results

Figure 2 shows the asymmetrically dynamic motion of a sub-micrometer dielectric particle in two lateral directions of an optical trap formed by a linearly polarized Gaussian beam. If the particle size (0.3 µm) is less than λ/2, where λ is the wavelength of the incident laser (λ=1.06 µm in our experiment), the particle is confined in an ellipse area on the transverse plane. In a linearly polarized trap, as shown in Fig. 2(a), the position fluctuation of the trapped particle in the parallel direction to the polarization of the incident laser field is larger than that that in the normal direction to the polarization. If the half-wavelength plate (λ/2) was replaced with a quarter-wavelength plate (λ/4), that is, the trapping laser is circularly polarized, the particle is confined in a circular area on the transverse plane, and the position fluctuation of the trapped particle on the transverse plane is symmetric, as shown in Fig. 2(b).

 

Fig. 2. The position fluctuations of 300nm-diameter polystyrene sphere (a) and (b), and correlation functions (c) and (d) in an optical trap with linear polarization and circular polarization, respectively.

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When the polarization direction of the incident laser was rotated with rotating the angle of the λ/2 plate, the axis of the position ellipse was rotated following the rotation of the polarization direction. Fig. 3 shows the mass center fluctuations of a 0.3-µm bead for the incident beam in different polarization directions at 0⁰, 45⁰, 90⁰, and 135⁰, respectively. The laser power was kept at 25 mW in these measurements. The diameter of the bead was well smaller than λ/2 and the position of the bead in the trap was confined in an ellipse area. The confinement of the Brownian motion in the parallel direction to the laser polarization is larger than that in the perpendicular direction, which indicates that k_‖>k_⊥.

It should be noted that this polarization-dependent asymmetry in the dynamic motion of a sub-micrometer transparent particle by a linearly polarized fundamental Gaussian beam is different from that of an optical spanner, in which a circularly polarized doughnut laser beam transfers its angular momentum to a trapped particle. In the experiment, we found that if the size of the polystyrene sphere is larger than or comparable to laser wavelength (such as for 1.0-µm sphere), the dynamic motion in the linearly polarized trap becomes almost symmetry in transverse plane.

 

Fig. 3. The mass center position fluctuations of a 300nm-diameter sphere in a linear-polarized trap as the laser polarization direction is rotated at (a) 0⁰, (b) 45⁰, (c) 90⁰, and (d) 135⁰. The laser power was kept at 25 mW.

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4. Theoretical analysis and discussions

The observed polarization-dependent asymmetry in dynamic motion and correlation functions can be understood in a theoretical frame based on two-dimensional Langevin equations that describe the random motion of a particle in an external potential. The equations of motion for the trapped particle in x-y directions are

where x and y are the displacement coordinates of the trapped particle in the parallel and perpendicular directions to the laser polarization, respectively, k_⊥ and k_‖ are corresponding force constants of the laser trap, f₁(t) and f₂(t) the random force components in x and y directions, and γ=6πηa the friction constant of the colloidal particle with radius a due to the viscosity of the surrounding fluid. f_c(t) is the induced random force of the laser electric field on the electrically charged colloidal particle. The random force components satisfy the following correlations:

and

The radiation force acting on a sub-micrometer Rayleigh particle whose diameter d is well less than λ includes the scattering force (F_scat=n_bP_scat/c) and the gradient force (F_grad =-n_bα∇E²/2), where E is the electric field of the incident laser beam, P_scat the power scattered, α the polarizability of the particle, and n_b the index of refraction of the surrounding solvent medium [1]. The trap potential originates from the gradient force. For an incident Gaussian beam, the intensity profile ${\mathrm{E}}_{\text{inc}}^2$=I₀exp{-(x²+y²)/2${\mathrm{w}}_0^2$-z²/2${\mathrm{w}}_{\mathrm{z}}^2$} is symmetric in the lateral direction and, therefore, the x and y components of the gradient force F_grad are equal (F_x=F_y), where w₀ and w_z are the waists of the focused laser beam in the lateral and axial directions, respectively, and ω the light frequency. The symmetric gradient force also applies to the Mie particles whose diameter is much larger than the wavelength by using a ray optics analysis [1]. Therefore, the trap force constants are equal, k_⊥=k_‖, in Eqs. (2) and (3).

In water, the colloidal particles (polystyrene spheres of 0.3-µm diameter), have an effective electrical charge of Z*~1.4×10³ e. For a focused laser beam of 0.1 MW/cm², the instantaneous electric field at the focus or the trap center is about 4.3×10³ V/cm. Therefore, the instantaneous Coulomb force that is in the parallel direction to the laser polarization is ~96 pN, which is much larger than the trapping force (-kx) and random forces f₁(t) or f₂(t) in magnitude. Although the motion of the colloidal particle cannot follow the rapid changing of the electric force at the light frequency, the electrical charged particle tends to move along the electric field direction due to this strong electric force. On the other hand, the charged particle is in Brownian motion, which can be modeled approximately by a random walk in which the particle executes large number of jumps per second and each jump with a distance l. When the particle jumps, the ions in the double layer will be displaced by a distance of order l in a time t_i=l ²/(2D_i) to relax back to establish a new steady-state ion distribution around the jumped charged particle, where Di is the diffusion constant of ion i. During this process, an instantaneous electric dipole will be created, which equals the product of the effective charge Z* and the displacement from the center of the positive ion double layer to the center of the colloidal particle. Apparently, this dynamic electric dipole is random both in orientation and in amplitude. Although the time-average laser force on this random dipole is zero, the strong electric field at the focus of the laser beam tends to align this random dipole in the laser polarization direction. Therefore, we can use a random electric force f_c(t) to include the effect of the direct electric force and ionic double layer on the dynamic motion of the charged colloid particle in the optical trap. Although the time average of the electric force is zero, <f_c(t)>=0, its fluctuation or correlation <f_c(t)f_c(t’)> is significant comparing to the Brownian random force f₁(t) and f₂(t).

Since this random electric force is parallel to the laser polarization, it forces the colloidal sphere moving in a larger average displacement than that in the perpendicular direction. Therefore, the electrically charged particle is confined to move in an ellipse area on the transverse plane. As the laser polarization is rotated, the long axis of this ellipse area will rotate accordingly. A numerical solution of Eqs. (2) and (3) gives the instantaneous displacement (x, y) at any time t and therefore can provide the trajectory of the dynamic motion of the colloidal particle in the optical trap. Experimentally, sub-micron-size macromolecules such as intralipid particle (d~100 nm) trapped in a linearly polarized optical trap also show similar asymmetry in their dynamic motion.

5. Conclusion

We have measured the dynamic motion of a nano-sized colloidal particle (polystyrene sphere of 300nm-diameter) in an optical trap formed by a linearly or a circularly polarized laser beam in the lateral direction. The experimental results show that in a linearly polarized optical trap, the position’s fluctuation of the trapped nano-sized particle in the parallel direction to the laser polarization is significantly larger than that in the normal direction. In a circularly polarized trap, this asymmetry in transverse dynamic motion disappears. The asymmetric dynamic motion in the linearly polarized optical trap is attributed to the random electric force induced by the laser field on the electrically charged nanometer particle. A theoretical analysis was developed to explain our experimental findings.