Dynamics of a Kerr Nanoparticle in a Single Beam Optical Trap

1. Introduction

Single beam optical traps also known as optical tweezers, are versatile optical tools for controlling precisely the movement of optically-small particles. Single-beam trapping was first demonstrated with visible light (514 nm) in 1986 to capture and guide individual neutral (nonabsorbing) particles of various sizes (Ashkin et. al., 1986). Optical traps were later used to orient and manipulate irregularly shaped microscopic objects such as viruses, cells, algae, organelles, and cytoplasmic filaments without apparent damage using an infrared light (1060 nm) beam (Ashkin, 1990). They were later deployed in a number of exciting investigations in microbiological systems such as chromosome manipulation (Liang et.al., 1993), sperm guidance in all optical in vitro fertilization (Clement-Sengewald et.al.,1996) and force measurements in molecular motors such single kinesin molecules (Svoboda and Block, 1994) and nucleic acid motor enzymes (Yim et.al., 1995). More recently, optical tweezer has been used in single molecule diagnostics for DNA related experiments (Koch et.al., 2002). By impaling the beads onto the microscope slide and increasing the laser power, it was tested that the bead could be "spot-welded" to the slide, leaving the DNA in a stretched state- a technique was used in preparing long strands of DNA for examination via optical microscopy.

Researchers continue to search for ways to the capability of optical traps to carry out multi-dimensional manipulation of particles of various geometrical shapes and optical sizes (Grier, 2003, Neuman & Block, 2004). Efforts in optical beam engineering were pursued to generate trapping beams with intensity distributions other than the diffraction-limited beam spot e.g. doughnut beam (He et.al., 1995, Kuga et.al., 1997), helical beam (Friese et.al., 1998), Bessel beam (MacDonald et.al., 2002). Multiple beam traps and other complex forms of optical landscapes were produced from a single primary beam using computer generated holograms (Liesener et.al., 2000; Curtis et.al., 2002, Curtis et.al., 2003) and programmable spatial light modulators (Rodrigo et.al., 2005, Rodrigo et.al., 2005).

Knowing the relationship between characteristics of the optical trapping force and the magnitude of optical nonlinearity is an interesting subject matter that has only been lightly investigated. A theory that accurately explains the influence of nonlinearity on the behavior of nonlinear particles in an optical trap would significantly broaden the applications of optical traps since most materials including many proteins and organic molecules, exhibit considerable degrees of optical nonlinearity under appropriate excitation conditions (Lasky, 1997, Clays et.al., 1993, Chemla & Zyss, 1987, Prasad & Williams, 1991, Nalwa & Miyata, 1997). One possible reason for the apparent scarcity of published studies on the matter is the difficulty in finding a suitable strategy for computing the intensity-dependent refractive index of the particle under illumination by a focused optical beam.

We have previously studied the dynamics of a particle in an optical trap that is produced by a single tightly focused continuous-wave (CW) Gaussian beam in the case when the refractive index n ₂ of the particle is dependent on the intensity I (Kerr effect) of the interacting linearly polarized beam according to: n ₂ = n ₂ ⁽⁰⁾ + n ₂ ⁽¹⁾ E*E, where n ₂ ⁽⁰⁾ and n ₂ ⁽¹⁾ I are the linear and nonlinear components of n ₂ , respectively. We have calculated the (time-averaged) optical trapping force that is exerted by a focused TEM₀₀ beam of optical wavelength on a non-absorbing mechanically-rigid Kerr particle of radius a in three different value ranges of the size parameter : (1) = 2a/ >>100 geometric optics (Pobre & Saloma, 1997), (2) 100 Mie scattering (Pobre & Saloma, 2002), and (3) << 100 Rayleigh scattering regime (Pobre & Saloma, 2006, Pobre & Saloma, 2008).

Here we continue our effort to understand the characteristics of the (time-averaged) optical trapping force F_trap that is exerted on a Kerr particle by a focused CW TEM₀₀ beam in the case when a 50/. A nanometer-sized Kerr particle (bead) exhibits Brownian motion as a result of random collisions with the molecules in the surrounding liquid. The Brownian motion is no longer negligible and has to be into account in the trapping force analysis. The characteristics of the trapping force are determined as a function of particle position in the propagating focused beam, beam power and focus spot size, ₀ , a, and relative refractive index between the nanoparticle and its surrounding medium. The behavior of the optical trapping force is compared with that of a similarly-sized linear particle under the same illumination conditions.

The incident focused beam polarizes the non-magnetic Kerr nanoparticle (a << ) and the electromagnetic (EM) field exerts a Lorentz force on each charge of the induced electric dipole (Kerker, 1969). We derive an expression for F_trap in terms of the intensity distribution and the nanoparticle polarizability = (n ₁ , n ₂ ), where n ₂ and n ₁ are the refractive index of the Kerr nanoparticle and surrounding medium, respectively. Optical trapping force (F_trap) has two components, one that accounts for the contribution of the field gradient and the other from the light that is scattered by the particle. The two-component approach for computing the magnitude and direction of F_trap was previously used on linear dielectric nanoparticles in arbitrary electromagnetic fields (Rohrbach & Steltzer, 2001). We also mention that the calculation of the intensity distributions near Gaussian beam focus is corrected up to the fifth order (Barton & Alexander, 1989).

In the next section, we will show the equation of the motion of a Kerr nanoparticle near the focus of a single beam optical trap in a Brownian environment. Simulation results will be presented and discussed in detail for other sections.

2. Theoretical framework

A linearly polarized Gaussian beam (TEM₀₀ mode) of wavelength , is focused via an objective lens of numerical aperture NA and allowed to propagate along the optical z-axis in a linear medium of refractive index n ₁ (see Fig 1). The beam radius _o at the geometrical focus (x = y = z = 0) is: _o = /(2NA).

Figure 1.

Nonlinear nanoparticle of radius a and refractive index n ₂ is located near the focal volume of a tightly-focused Gaussian beam of wavelength >> a and beam focus radius _o . Gaussian beam propagates in a linear medium of index n ₁ . Nanoparticle center is located at r(x, y, z) from the geometrical focus at r(0, 0, 0). Enlarged figure in the focal volume shows Kerr nanoparticle undergoing Brownian motion near the focus.

The focused beam interacts with a Kerr particle of radius a 50/. The refractive index n ₂ (r) of the Kerr particle is given by: n ₂ (r) = n ₂ ⁽⁰⁾ + n ₂ ⁽¹⁾ I(r), where I(r) = E*(r)E(r) is the beam intensity at particle center position r = r(x, y, z) from the geometrical focus at r = 0 which also serves as the origin of the Cartesian coordinate system. Throughout this paper, vector quantities represented in bold letters.

The thermal fluctuations in the surrounding medium (assumed to be water in the present case) become relevant when the particle size approaches the nanometer range. We consider a Kerr nanoparticle that is located at r above the reference focal point in the center of the beam waist ₀ that is generated with a high NA oil-immersed objective lens of an inverted microscope – the focused beam propagates in the upward vertical direction (see inset Fig. 1).

The dynamics of the Kerr nanoparticle as it undergoes thermal diffusion can be analyzed in the presence of three major forces: (1) Drag force, F_drag(dr/dt) = F_drag, that is experienced when the particle is in motion, (2) Trapping force F_trap(r), which was derived in (Pobre & Saloma, 2006), and (3) time-dependent Brownian force F_fluct(t) = F_fluct, that arise from thermal motion of the molecules in the liquid. The Kerr nanoparticle experiences a net force F_net(r, t) = F_net, that can be expressed in terms of the Langevin equation as:

$$\begin{array}{l}{F}_{\text{net}}(r,\text{t})=F_{\text{drag}}(\dot{r})+F_{\text{trap}}(r)+F_{\text{fluct}}(\text{t})\\ \text{m}\ddot{r}=-\gamma \dot{r}+F_{\text{trap}}(r)+F_{\text{fluct}}(\text{t})\end{array}$$ E1

where: F_drag = - dr/dt, and is the drag coefficient of the surrounding liquid. According to Stokes law, = 6a, where is the liquid viscosity. While the optical trapping force or optical trapping force, F_trap(r), on the Kerr nanoparticle was shown to be (Pobre & Saloma, 2006):

$$F_{\text{trap}}(r)=\left(\frac{2\pi {\text{ n}}_{\text{1}}{\text{ a}}^3}{\text{c}}\right){\left(\frac{{\left\{\frac{{\text{n}}_{\text{2}}^{\text{(0)}}+{\text{n}}_{\text{2}}^{\text{(0)}}\text{I(r)}}{{\text{n}}_{\text{1}}}\right\}}^2-1}{{\left\{\frac{{\text{n}}_{\text{2}}^{\text{(0)}}+{\text{n}}_{\text{2}}^{\text{(0)}}\text{I(r)}}{{\text{n}}_{\text{1}}}\right\}}^2+2}\right)}^2\nabla \text{I(r)}+\left(\frac{8\pi {\text{ n}}_{\text{1}}}{3\text{c}}\right){(\left|k\right|\text{ a })}^4{\text{a}}^2{\left(\frac{{\left\{\frac{{\text{n}}_{\text{2}}^{\text{(0)}}+{\text{n}}_{\text{2}}^{\text{(0)}}\text{I(r)}}{{\text{n}}_{\text{1}}}\right\}}^2-1}{{\left\{\frac{{\text{n}}_{\text{2}}^{\text{(0)}}+{\text{n}}_{\text{2}}^{\text{(0)}}\text{I(r)}}{{\text{n}}_{\text{1}}}\right\}}^2+2}\right)}^2\text{I(r)}$$ E2

Equation (2) reveals that F _trap consists of two components. The first component represents the gradient force and depends on the gradient of I(r) and it is directed towards regions of increasing intensity values. The second component represents the contribution of the scattered light to F _trap . The scattering force varies with I(r) and it is in the direction of the scattered field. Hence, the relative contribution of the scattering force to F _trap is weak for a particle that scatters light in an isotropic manner.

The Gaussian beam has a total beam power of P (Siegman, 1986) and its intensity distribution I(r) near the beam focus is calculated with corrections introduced up to the fifth-order (Barton & Alexander, 1989). Focusing with a high NA objective produces a relatively high beam intensity at z = 0, which decreases rapidly with increasing |z| values. On the other hand, low NA objectives produce a slowly varying intensity distribution from z=0.

The molecules of the surrounding fluid affect significantly on the mobility of the Kerr nanoparticle since their sizes are comparable. As a result, the Kerr nanoparticle moves in a random manner between the molecules and exhibits the characteristics of a Brownian motion. The associated force can be generated via a white-noise simulation since it mimics the behavior of the naturally occurring thermal fluctuations of a fluid. The assumption holds when both the liquid and the Kerr nanopartilcle are non-resonant with . Localized (non-uniform) heating of the liquid is also minimized by keeping the average power of the focused beam low for example with a femtosecond laser source that is operated at high peak powers and relatively low repetition rate.

3. Optical trapping potential

As previously discussed, the Kerr nanoparticle of mass m and 2a/ 100 and a << , exhibits random (Brownian) motion in the liquid (Rohrbach & Steltzer, 2002, Singer et.al., 2000). The thermal fluctuation probability increases with the temperature T of the liquid. To determine the dynamics of a Kerr nanoparticle near the focus of a single beam optical trap, we first determine the potential energy V(r) of the optical trap near the beam focus, which can be characterized in terms of F_trap. The potential V(r) as a function of the optical trapping force from all axes (in this case along the x, y, and z axes) is given by:

$$\begin{array}{l}\text{V(r)}=-{\displaystyle \underset{r_0}{\overset{r_f}{\int }}{F}_{\text{trap}}(}r)dr\\ =-{\displaystyle \underset{x_0}{\overset{x_f}{\int }}{F}_{\text{trap},x}(}r)dx-{\displaystyle \underset{y_0}{\overset{y_f}{\int }}{F}_{\text{trap},y}(}r)dy-{\displaystyle \underset{z_0}{\overset{z_f}{\int }}{F}_{\text{trap},z}(}r)dz\end{array}$$ E3

where: F_trap,x, F_trap,y and F_trap,z are the Cartesian components of F_trap, and r₀(x₀, y₀, z₀; t₀) = r₀(t₀) and r_f(x_f, y_f, z_f; t_f) = r_f(t_f) are the initial and final positions of the nanoparticle. For a nanoparticle in the focal volume of a Gaussian beam, V(r) can be approximated as a harmonic potential since the magnitude of F_drag is several orders larger than that of the inertial force. Equation (1) then describes an over-damped harmonic motion that is driven by time-dependent thermal fluctuations.

A nanoparticle at location r(t) in the optical trap has a potential energy V(r) and a kinetic energy m|v|²/2 where v = v(t) is the nanoparticle velocity. The probability that the Kerr nanoparticle is found at position r(t), is described by a probability density function (r) = ₀ exp[-V(r)/k _B T], where ₀ is the initial probability density, T is the temperature of the surrounding medium, and k _B is the Boltzmann constant.

Figure 2 plots the potential energy (2a) of the optical trap and the corresponding time-dependent displacement trajectory (2b) of the Kerr nanoparticle (initial z position = 0.4 m) along the optical z-axis assuming a zero initial velocity and a room temperature condition of 3.1 k_bT background energy of the surrounding medium. The trajectory (in blue trace) can be ascribed as overdamped oscillations of the Kerr nanoparticle that arise from the complex interplay of three forces indicated in the Langevin’s differential equation. The oscillations

Figure 2.

a) Potential energy and probability density function along the z-axis with trapping input parameters: z_o=0, p=100mW, a=30nm, N.A.=1.2, =1.064m, n ₁ =1.33, n ₂ ⁽⁰⁾ =1.4, and n ₂ ⁽¹⁾ =1.8 x 10^-12m²/W. (b) Thermal diffusion of the Kerr nanoparticle along the z-axis with zero initial velocity at 0.4 m with a 3.1 k_bT ambient energy (T=300K) of the surrounding water (in red dashed line).

are caused by random collisions between the Kerr nanoparticle and the relatively-large molecules. The narrower confinement of the Kerr nanoparticle indicates a stiffer potential trap that is contributed by the effects of the nonlinear interaction between the Kerr nanoparticle and the tightly focused Gaussian beam.

Figure 3 presents the three-dimensional (3D) plots of the trapping potential that is created by a focused beam (NA = 1.2) in the presence of a linear and a Kerr particle. The potential wells are steeper along the x-axis than along the z-axis since a high NA objective lens produces a focal volume that is relatively longer along the z-axis. The potential well associated with a Kerr nanoparticle is deeper than that of a linear nanospshere.

Figure 3.

Three-dimensional plot of the trapping potential energy along the transverse plane for both linear and nonlinear nanosphere as the focused laser beam propagates from left to right of the z-axis with the following trapping parameters: z_o=0, p=100mW, a=30nm, N.A.=1.2, =1.064um, n₁=1.33, n ₂ ⁽⁰⁾ =1.4, and n ₂ ⁽¹⁾ =1.8 x 10^-12m²/W.

Under the same illumination conditions, a Kerr nanoparticle is captured more easily and held more stably in a single beam optical trap than a linear nanoparticle of the same size. A Kerr nanoparticle that is exhibiting Brownian motion is also confined within a much smaller volume of space around the beam focus as illustrated in 3D probability density of figure 4. The significant enhancement that is introduced by the Kerr nonlinearity could make the simpler single-beam optical trap into a viable alternative to multiple beam traps which are costly, less flexible and more difficult to operate.

Figure 4.

Probability density distributions of linear and nonlinear (Kerr) nanospheres in a single-beam optical trap at T = 300K where t = 100,000 iterations, P = 100mW, a = 5 nm, NA = 1.2, = 1.064 m, and n ₁ = 1.33: a) Location probability distribution of linear (n ₂ = n ₂ ⁽⁰⁾ ) and b) Kerr nanoparticle (n ₂ ⁽⁰⁾ = 1.4, n ₂ ⁽¹⁾ = 1.8 x 10^-12 m²/W). Initially (t = 0), the nanoparticle is at rest at z = 0.5 m.

4. Parametric analysis of the optical trapping force between linear and nonlinear (Kerr) nanoparticle

To better understand the underlying mechanism on how Kerr nonlinearity affects the trapping potential, let us perform a parametric analysis on how optical trapping force changes with typical trapping parameters on both linear and nonlinear (Kerr) nanoparticle.

The optical trapping force F_trap(r) that is described by Eq (2) was calculated using Mathematica Version 5.1 application program. Figure 5a presents the contour and 3D plots of F_trap(r) at different locations of the linear nanoparticle (n ₂ = 1.4, a = 5 nm, = 1.062 m, NA = 1.2) while Figure 5b shows the contour and 3D plots of F_trap(r) at different locations of the Kerr nanoparticle (n ₂ ⁽⁰⁾ = 1.4, n ₂ ⁽¹⁾ = 1.8 x 10^-11 m²/W, a = 5 nm, = 1.062 m, NA = 1.2). The n ₂ ⁽¹⁾ value is taken from published measurements done with photopolymers which are materials that exhibit one of the strongest electro-optic Kerr effects (Nalwa & Miyata, 1997). Also shown is the contour plot of F_trap(r) for the case of a linear nanoparticle (n ₂ ⁽⁰⁾ = 1.4, a = 5 nm) of the same size.

For values of z > 0, F_trap is labeled negative (positive) when it pulls (pushes) the nanoparticle towards (away from) r = 0. For z ≤ 0 the force is positive (negative) when it pushes (pulls) the nanoparticle towards (away from) the beam focus at r = 0. For both linear and nonlinear nanoparticles, the force characteristics are symmetric about the optical z-axis but asymmetric about the z = 0 plane. The asymmetry of the force is revealed only after the fifth-order correction is applied on the intensity distribution of the tightly focused Gaussian beam. The strongest force magnitude happens on the z-axis and it is 30% stronger in the case of the Kerr nanoparticle.

The stiffness of the optical trap may be determined by taking derivative of F_trap(r) with respect to r. Figure 6b plots the stiffness at different locations of the Kerr nanoparticle. The stiffness distribution features a pair of minima at r = (x² + y²)^1/2 ≈ 0.1 micron with a value of -25 x 10^-12 N/m. Also presented in Fig 6a is the force stiffness distribution for the case of a

Figure 5.

Optical trapping force at different locations of both linear and Kerr nanoparticle (n ₂ ⁽⁰⁾ = 1.4, n ₂ ⁽¹⁾ = 1.8 x 10^-11 m²/W) where r = (x² + y²)^1/2. Parameter values common to both nanoparticles: P = 100 mW, a = 5 nm, NA = 1.2, = 1.064 microns, and n ₁ = 1.33. The focused beam propagates from left to right direction. In all cases, F _trap = 0 at r(x, y, z) = 0.

linear nanoparticle exhibits a similar profile but a lower minimum value of -18 x 10^-12 N/m at r ≈ 0.1 micron. The Kerr nanoparticle that is moving towards r = 0, experiences a trapping force that increases more rapidly than the one experienced by a linear nanoparticle of the same size. Once settled at r = 0, the Kerr nanoparticle is also more difficult to dislodge than its linear counterpart.

Figure 7a plots the behavior of F_trap at different axial locations of a linear nanoparticle (n₂ = 1.4) with a(nm) = 50, 70, 80, 90 and 100. In larger Kerr nanoparticles (a > 50 nm), the scattering force contribution becomes significant and the location of F_trap(r) = 0 shifts away from z = 0 and towards z > 0. Our results are consistent with those previously reported with linear dielectric nanoparticles (Rohrback and Steltzer, 2001, Wright et.al., 1994).Figure 7b plots the behavior of F_trap(r) at different axial locations of a bigger Kerr nanoparticle with a(nm) = 50, 70, 80, 90 and 100. The maximum strength of F_trap(r) increases with a. For a < 50 nm, F_trap(r) = 0 at z = 0 since F_trap(r) is contributed primarily by the gradient force. For larger Kerr nanoparticles, the relative contribution of the scattering force becomes more significant and the location where F_trap(r) = 0 is shifted away from z = 0 and towards the direction of beam propagation.Figure 8a plots the behavior of F _trap (r) as a function of the objective NA (0.4 ≤ NA ≤ 1.4) for a Kerr nanoparticle [n ₂ ⁽⁰⁾ = 1.4, n ₂ ⁽¹⁾ = 1.8 x 10^-11 m²/W, P = 100 mW, a = 5 nm) that is located at r(0, 0, 0.5 micron). Also plotted is the behavior of F _trap (r) with NA for a linear nanoparticle of the same size and initial beam location. Both the Kerr and the linear nanoparticle

Figure 6.

Optical trapping force stiffness of optical trap at different locations of both linear (n ₂ ⁽⁰⁾ = 1.4) and Kerr nanoparticle (n ₂ ⁽⁰⁾ = 1.4, n₂ ⁽¹⁾ = 1.8 x 10^-12 m²/W) where r = (x² + y²)^1/2. Common parameter values: P = 100 mW, a = 5 nm, NA = 1.2, = 1.064 microns, and n ₁ = 1.33.

Figure 7.

Optical trapping force at different axial locations of: a) linear (n ₂ = n ₂ ⁽⁰⁾ = 1.4), and b) Kerr (n ₂ ⁽⁰⁾ = 1.4, n₂ ⁽¹⁾ = 1.8 x 10^-12 m²/W) nanoparticle of radius a(nm) = 50, 70, 80, 90 and 100. Common parameter values: P = 100 mW, NA = 1.2, = 1.064 microns, and n ₁ = 1.33.

experience a trapping force that pulls them towards r = 0. The effect of the Kerr nonlinearity which is to increase the strength of F _trap (r), becomes more significant at NA > 1. At NA = 1.4, the force magnitude on the Kerr nanoparticle approximately twenty percent stronger than that experienced by the linear nanoparticle. The nonlinear effect is negligible in low NA focusing objectives (NA < 0.6). For the Kerr nanoparticle, the dependence of the force strength with NA is accurately described by a fourth order polynomial.

Figure 8.

a) Optical trapping force on Kerr nanoparticle (solid line; n ₂ ⁽⁰⁾ = 1.4, n ₂ ⁽¹⁾ = 1.8 x 10^-12 m²/W) as a function of objective NA. Also plotted is the corresponding force (dotted line) on a linear nanoparticle with n ₂ = n ₂ ⁽⁰⁾ = 1.4. Common parameter values: z = 0.5 micron, P = 100 mW, a = 5 nm, = 1.064 microns, and n ₁ = 1.33. For the Kerr nanoparticle the curve is accurately described by: F_trap = 1311.65(NA)⁴ – 3959.965(NA)³ + 3656.265(NA)² – 1388.114NA + 188.105. (b) Optical trapping force on non-absorbing Kerr (solid line; n ₂ ⁽¹⁾ = 1.8 x 10^-12 m²/W) and linear (n ₂ = n ₂ ⁽⁰⁾ = 1.4) nanoparticle versus n ₂ ⁽⁰⁾ , at = 1.064 microns. Also plotted (circles) is force difference (F_nl – F_l) as a function of n ₂ ⁽⁰⁾ .

Figure 8b shows the behavior of F _trap (r) as a function of n₂ ⁽⁰⁾ for a Kerr nanoparticle that is located at r(0, 0, 0.5 micron) in a surrounding medium with index n₁ = 1.33. Also plotted is the behavior of F _trap (r) with n₂ ⁽⁰⁾ for a linear nanoparticle (n ₂ = n ₂ ⁽⁰⁾ ) of the same size and initial beam location. The F _trap (r) profiles are similar for both the Kerr and linear nanoparticles. At n ₂ ⁽⁰⁾ > 1.1, the Kerr nanoparticle experiences a negative (trapping) force that pulls it towards from r = 0. The trapping threshold is less than n ₂ ⁽⁰⁾ = 1.33 because of the additional contribution of the nonlinear (Kerr) term n ₂ ⁽¹⁾ I(r). For the linear nanoparticle, trapping is possible at a higher value of n ₂ ⁽⁰⁾ > 1.33. Also plotted is the difference between the forces that are experienced by the two nanoparticles. The difference between the two trapping forces is highest near n₂ ⁽⁰⁾ = 1. The difference decreases with increasing n₂ ⁽⁰⁾ value since the contribution of the Kerr term which has been held constant, becomes relatively small.Figure 9 shows the dependence of F_trap(r) with for a non-resonant Kerr nanoparticle at r(0, 0, 0.5 micron) in the range: 400 ≤ (nm) ≤ 1000. Also presented is the behavior of F_trap(r) for a linear nanoparticle of the same size and initial beam location. For a given P and NA value, the magnitude of F_trap(r) increases nonlinearly with decreasing for both nanoparticles. However, the increase in the trapping force strength with is more rapid for the Kerr nanoparticle. In practice, is selected to avoid absorption by the nanoparticle and the surrounding liquid. Absorption could significantly heat up the nanoparticle and change its optical and mechanical properties. It can also lead to rapid evaporation of the surrounding liquid. In both cases, absorption reduces the efficiency of the optical trap.

Figure 9.

Optical trapping force on non-absorbing Kerr nanoparticle (solid line, n₂ ⁽⁰⁾ = 1.4, n₂ ⁽¹⁾ = 1.8 x 10^-12 m²/W) as a function of wavelength . Also plotted (dotted line) is the force on a linear nanoparticle (n₂ = n₂ ⁽⁰⁾ = 1.4). Other parameter values: P = 100 mW, a = 5 nm, NA = 1.2, z = 0.5 micron, and n ₁ = 1.33.

Figure 10a plots the dependence of F_trap(r) with beam power P for a Kerr nanoparticle at r(0, 0, 0.5 micron). Also presented is the behavior of F_trap(r) for a linear nanoparticle of the same size and initial beam location. For the linear nanoparticle, the trapping force strength is directly proportional to P. For the Kerr nanoparticle, the trapping force strength increases more quickly (quadratically) with P for the Kerr nanoparticle. Figure 8b reveals that the force strength increases at a faster rate as the Kerr nanoparticle gets bigger.

Figure 10.

a) Optical trapping force on Kerr (filled circles; a = 5 nm) and linear nanoparticle (circles; a = 5 nm) as a function of beam power P, and (b) Force versus P for different radii of Kerr nanoparticle. Common parameter values: NA = 1.2, z = 0.5 micron), n₂ = n₂ ⁽⁰⁾ = 1.4, n₂ ⁽¹⁾ = 1.8 x 10^-12 m²/W, and n ₁ = 1.33. In (a) the force F_trap acting on the Kerr nanoparticle is accurately described by: F_trap = 0.006P² –2.742P + 0.052.

Figure 11, plots the trapping potential V(r) at different axial locations of a nanoparticle (a = 5 nm, = 1.064 microns, P = 100 mW) in the absence (n ₂ = n ₂ ⁽⁰⁾ ) and presence (n ₂ ⁽¹⁾ = 1.8 x 10^-11 m²/W) of Kerr nonlinearity. Compared to its linear counterpart, the Kerr nanoparticle is subjected to a V(r) that is significantly deeper and narrower. From the previous plots, Figure 5a and 5b plot the movement of a linear and Kerr nanoparticle respectively, when they exhibit Browning motion in the single beam optical trap. The Kerr nanoparticle is released from rest at r(0, 0, 0.5 m) and it is pulled towards r = 0 and confined to move within a region of about 0.2 m radius, that is centered at r = 0 (Fig 5b). A linear nanoparticle that is released at r(0, 0, 0.5 m) is constrained to move within a larger region of about 0.5 m radius (Fig 5a).

Figure 11.

Optical trapping potential in a single-beam optical trap at T = 300K where t = 100,000 iterations, P = 100mW, a = 5 nm, NA = 1.2, = 1.064 m, and n ₁ = 1.33: Kerr nanoparticle (n ₂ ⁽⁰⁾ = 1.4, n ₂ ⁽¹⁾ = 1.8 x 10^-12 m²/W). Initially (t = 0), the nanoparticle is at rest at z = 0.5 m.

Figure 12.

Optical trapping potential at different axial locations of: a) linear (n ₂ = n ₂ ⁽⁰⁾ = 1.4), and b) Kerr (n ₂ ⁽⁰⁾ = 1.4, n ₂ ⁽¹⁾ = 1.8 x 10^-12 m²/W) nanoparticle of radius a(nm) = 50, 70, 80, 90 and 100. Common parameter values: P = 100 mW, NA = 1.2, = 1.064 microns, and n ₁ = 1.33.

Figure 12a presents the V(z) profile at different z-locations of a linear nanoparticle with a(nm) = 50, 70, 80, 90 and 100. For comparison, Figure 12b shows the behavior of V(r) at different z-locations of a Kerr nanoparticle ( = 1.064 microns, P = 100 mW, n₂ ⁽¹⁾ = 1.8 x 10^-11 m²/W) with radius a(nm) = 50, 70, 80, 90 and 100. With respect to z = 0, the V(z) profile becomes asymmetric with increasing a for a > 50 nm. The distortion is caused by the increasing contribution of the scattering force to the net force F_trap(r). It shifts the location of potential minimum away for z = 0, as well as lowers the escape threshold of a trapped Kerr nanoparticle in the direction of beam propagation. Optical trapping potential trends are similar except for differences in strengths, the V(z) profiles evolve in a similar manner with increasing nanoparticle size for both linear and nonlinear case.

5. Enhancement of single-beam optical trap due to Kerr nonlinearity

Our simulation results indicate that the performance of the single-beam optical trap is enhanced by the Kerr effect. For the same focused beam, a Kerr nanoparticle (n ₂ ⁽⁰⁾ = 1.4, n ₂ ⁽¹⁾ = 1.8 x 10^-11 m²/W, P = 100 mW, NA = 1.2, = 1.064 microns, n ₁ = 1.33) is subjected to a stronger trapping force than a linear nanoparticle (n ₂ = 1.4,) of the same size (see Figs 7 - 10). The force magnitude increases rapidly as the nanoparticle approaches geometrical focus at r = 0 (Fig 8) especially along the optical z-axis. At the minimum of the trapping potential V(r), a Kerr nanoparticle encounters a higher escape threshold and therefore needs a greater amount of kinetic energy to escape from the optical trap (see Fig 11). Under the same illumination conditions, a Kerr nanoparticle that is exhibiting Brownian motion, is confined to move to within a much smaller region around r = 0, that a linear nanoparticle of the same size (see Fig 5).

The optical trapping force F_trap that is exerted on a Kerr nanoparticle with a ≤ 50 nm = /21.3, is contributed primarily by the gradient force component. In such cases, F_trap = 0 at z = 0 (see Figs 8) and V(z) is symmetric about z = 0 (Figs 12 – 13). At a = 5 nm, we found that the maximum strength of the gradient force is about three orders of magnitude larger than that of the scattering force. The axial location where F_trap = 0 is shifted away from z = 0 and towards the general direction of the beam propagation, when the contribution of the scattering force component becomes comparable (see Fig 11). The corresponding V(z) profile becomes asymmetric with a lower escape barrier along the direction of beam propagation (see Fig 13). Such instances occur with larger Kerr nanoparticles (a > /21.3).

Except for differences in their relative magnitudes, the axial profiles of F_trap exhibit the similar characteristics with increasing nanoparticle size for the both the nonlinear and linear case. Our results indicate that the index increase that is introduced by the Kerr effect, does not affect the ability of a small Kerr nanoparticle (a ≤ /21.3) to scatter light in an isotropic manner - the increase in n₂ is uniform distributed in the nanoparticle. The gradient force contribution to F_trap becomes significant when the non-absorbing nanoparticle scatters light in an anisotropic manner.

Figure 8a illustrates that the enhancement that is gained from the Kerr effect in trapping a non-resonant nanoparticle, is realized only with high NA focusing objectives (NA > 0.6). The strength of F_trap becomes stronger at shorter values (see Fig.10). The increase is faster for the Kerr nanoparticle due to the dependence of its refractive index with I(r) – the force strength increases quadratically with . We note that the strong dependence of F_trap with is not observed in larger Kerr nanoparticles especially in the regime of > 100 and a >> ) (Pobre & Saloma, 1997, Pobre & Saloma, 2002).

The optical trapping force increases rapidly with beam power P for the same NA and values (see Fig 10a). For a linear nanoparticle, the force strength is directly proportional to P. For a Kerr nanoparticle, the relationship of the force strength with P is nonlinear - the Kerr effect permits the use of low power light sources that tend to be less costly to acquire and maintain. Trapping at low beam powers also minimizes the optical heating of the surrounding medium and even the nanoparticle itself. Reductions in unwanted thermal effects are vital in the manipulation and guidance of biological samples.

6. Summary and future prospects

We have analyzed the optical trapping force F_trap that is exerted on a Kerr nanoparticle by a focused Gaussian beam when 2a/ 100 and a << . The optical trapping mechanism consists of two dominant optical forces representing the contribution of the field gradient and that of the EM field that is scattered by the nanoparticle. The contributions of the two force components become comparable for nanoparticles with a > /21.3. The gradient force contribution is more dominant with smaller non-absorbing nanoparticles such that F_trap = 0 at the beam focus r = 0. The Brownian motion of the Kerr nanoparticle has an over-damped harmonic motion enveloped by white noise function due to thermal fluctuations generated by moving molecules defined by the background energy of 3.1 k_bT of the surrounding fluid. Confinement of the Kerr nanoparticle depends on the nonlinear refractive index of the nanoparticle as shown in the widths of the probability density of the Kerr nanoparticle.

Under the same illumination conditions, a Kerr nanoparticle is captured more easily and held more stably in a single beam optical trap than a linear nanoparticle of the same size. A Kerr nanoparticle that is exhibiting Brownian motion is also confined within a much smaller volume of space around the beam focus. The significant enhancement that is introduced by the Kerr nonlinearity could make the simpler single-beam optical trap into a viable alternative to multiple beam traps which are costly, less flexible and more difficult to operate.

Kerr nonlinearity enhances the performance of a single beam trap by increasing the magnitude of the trapping force. Its permits the trapping of nonlinear nanoparticles with n ₂ ⁽⁰⁾ values that are less than the index n₁ of the surrounding liquid and at lower NA values and optical beam powers. Low NA focusing objectives and low power laser sources are relatively inexpensive and are less likely to cause irreversible thermal damage on the sample and the surrounding medium.

Localized (non-uniform) heating of the liquid is also minimized if the average power of the focused beam is kept low using a femtosecond laser source with high peak powers and relatively low repetition rate.

Kerr nanoparticle can be an alternative probe handler when applied to photonic force microscope configuration for the imaging of hollow microbiological structures.

Acknowledgments

The authors are grateful for the financial support provided by University Research Coordination Office of De La Salle University (DLSU) and the University of the Philippines Diliman.