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
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
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
Here we continue our effort to understand the characteristics of the (time-averaged) optical trapping force Ftrap that is exerted on a Kerr particle by a focused CW TEM00 beam in the case when
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 Ftrap in terms of the intensity distribution and the nanoparticle polarizability = (n
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
The focused beam interacts with a Kerr particle of radius a 50/. The refractive index n
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
The dynamics of the Kerr nanoparticle as it undergoes thermal diffusion can be analyzed in the presence of three major forces: (1) Drag force, Fdrag(dr/dt) = Fdrag, that is experienced when the particle is in motion, (2) Trapping force Ftrap(r), which was derived in (Pobre & Saloma, 2006), and (3) time-dependent Brownian force Ffluct(t) = Ffluct, that arise from thermal motion of the molecules in the liquid. The Kerr nanoparticle experiences a net force Fnet(r, t) = Fnet, that can be expressed in terms of the Langevin equation as:
where: Fdrag = - 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, Ftrap(r), on the Kerr nanoparticle was shown to be (Pobre & Saloma, 2006):
Equation (2) reveals that F
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 Ftrap. 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:
where: Ftrap,x, Ftrap,y and Ftrap,z are the Cartesian components of Ftrap, and r0(x0, y0, z0; t0) = r0(t0) and rf(xf, yf, zf; tf) = rf(tf) 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 Fdrag 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
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.
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.
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 Ftrap(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 Ftrap(r) at different locations of the linear nanoparticle (n
For values of z > 0, Ftrap 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 Ftrap(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
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 Ftrap 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 Ftrap(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 Ftrap(r) at different axial locations of a bigger Kerr nanoparticle with a(nm) = 50, 70, 80, 90 and 100. The maximum strength of Ftrap(r) increases with a. For a < 50 nm, Ftrap(r) = 0 at z = 0 since Ftrap(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 Ftrap(r) = 0 is shifted away from z = 0 and towards the direction of beam propagation.Figure 8a plots the behavior of F
experience a trapping force that pulls them towards r = 0. The effect of the Kerr nonlinearity which is to increase the strength of F
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
The optical trapping force Ftrap that is exerted on a Kerr nanoparticle with a ≤ 50 nm = /21.3, is contributed primarily by the gradient force component. In such cases, Ftrap = 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 Ftrap = 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 Ftrap 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 Ftrap 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 Ftrap 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 Ftrap 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 Ftrap 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 Ftrap = 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 kbT 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
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.
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.
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