Laser light can exert forces on matter by exchanging momentum in form of radiation pressure and refraction. Although these forces are small, they are sufficient to trap and manipulate microscopic particles [Phys. Rev. Lett. 24, 156 (1970)]. In this paper, we study the optical trapping phenomena by using computer simulation to show a detailed account of the process of momentum exchange between a focused light and a microscopic particle in an optical trapping by use of the finite difference time domain method. This approach provides a practical routine to predict the magnitude of the exchanged momentum, track the particle in a trapping process, and determine a trapping point, where dynamic equilibrium happens. Here we also theoretically describe the transfer procedure of orbital angular momentum from a focused optical vortex to the particle.
©2004 Optical Society of America
Light can be described as photons with momentum of p = h/λ per photon, where p is the momentum, h is the Planck’s constant, and λ is the wavelength. The intensity (or energy density) of light is determined by the number of the photons passing through a given area per unit time and it can be calculated by the poynting vector S, and thus the momentum flux of light per unit area dA can be given by:
where P is the momentum of light, n is refractive index and c is the speed of light. The force F exerted on an object by light can be determined by:
where F is the total force acted on the object and ΔS represents the difference between the energy density flux through the unit area traveling into the object and coming out of the object. This shows that light exerts forces while it is refracted or reflected at the interface between two different media, and the optical force or torque acted on the particle within a laser trap results from the change in momentum of light due to refraction or absorption.
In optical tweezers experiment, it can be experimentally demonstrated that a laser beam can provide a finite force of the order of pico-Newton, which can trap particles of a wide range of sizes from about 50nm to 20μm. With the development of some simple trapping and manipulation technologies reported for applications such as laser trapping in various scientific fields , near field optical trapping  has become extensively attractive, and thus it has become very important to learn about the detailed mechanism of various momentum transfers from light to the object in a laser trapping. The use of laser trapping for quantitative research, especially using a trapped particle as a probe, requires accurate values of the optical forces in the trapping. An accurate quantitative theory for laser trapping will not only extend and calibrate the technique of optical force and torque measurement and trapped particle characterization, but also reveal the physical property of light and matter. The exchange of momentum in optical traps in Eq. (2) can be determined by the electromagnetic (EM) theory. The EM field computation involving inhomogeneous and arbitrary-shaped objects can be carried out conveniently by the finite-difference time-domain  (FDTD) method, which is a straightforward approach for the Maxwell’s equations. We first converted the optical trap space into meshes, and then used the FDTD to calculate the electric and magnetic fields of the optical traps in the spatial domain as a function of time steps. The momentum of light can be determined from S = E×H in the time domain . Although the FDTD is a time consuming method for an EM problem, there are many unique advantages for simulating optical trappings. For example, FDTD allows to analyze arbitrary shape scattering objects, and to easily induce a focused light source, which is very important because lots of optical trapping experiments were demonstrated within the focused light field generated from a high numerical aperture (N.A.) objective lens within a microscope. In addition, the FDTD can give a scattering field in time domain, which is very useful to study optical trapping phenomena in ultra-fast optics and near field optical microscopy.
In this paper, we present the FDTD simulations of the optical trapping in a focused single sinusoidal pulse wave (SSPW) and that in a focused optical vortex pulse beam (OVPB) with a helical phase. The former presents different actions of the SSPW to the particles with different optical properties, and here FDTD can describe the momentum exchange in time domain between light and microscopic particles. The latter is used to theoretically study the orbit angular momentum (OAM) transfer between the focused OVPB and the microscopic particle.
2. Geometry of the optical trapping model
In many cases, since a microscopic optical tweezers based optical trapping system always confined the object at the beam centre, which is usually on the optical axis, the transverse momentum (S x and S y, the components of S in x- and y-axis in Fig. 1) transfers are most important in an optical trapping within a microscope, and they are corresponding with the gradient force and the axial torque. It can be understood from the Maxwell’s equations that the transverse momentum is associated with focus or dispersion of light, and thus the optical trapping is generally observed in the focused light produced by an objective lens with high numerical aperture in a microscope. The field of the focused light can be described mathematically using a spherical wave going through the focal point. Fig. 1 shows the geometry of the model in the calculation field; the light source is set as a sinusoidal wave or a Gaussian pulse of light wave at wavelength of 632.8nm, and the focused field is simulated using a spherical wave induced by arranging the incident source on a given arc shape surface (depends on the expected N.A.) on the top of the focal point, which is located at the middle of the calculation region. The 3D calculation region consists of 120 × 120 × 165 Yee’s meshes over 3.3μm × 3.3μm × 4.5μm area in the x-, y- and z-direction of trapping space, respectively, and an outmost 10-mesh perfect match layer  is set as absorbing boundary. The temporal step size Δt is taken as 52.8 attosecond for stability criterion.
A sphere particle with refractive index n s is immersed in the water (green color, the refractive index n w = 1.33) and it is placed off the beam axis under the focal point. When light travels into the momentum transfer region (MTR), as shown in Fig. 1 on the emergent surface of the particle, with a diameter of two grids bigger than that of the particle, the momentum in this region can be calculated. We add up the light momentum passing through the MTR without and with the particle, and then compare the momentum change between the two cases. Due to conversation of momentum, the difference of light momentum in the two cases should be transferred to the particle. The reflective indices of the particle (n p) and the surrounding medium (n m) are important parameters for an effective laser trapping. It is well known that there exists great differences for the momentum transfer between light and the particle medium with a high refractive index (n p > n w, refractive index of water); or between light and the particle medium with a low refractive index (n p < n w); or between light and the particle medium with a high reflection coefficient at the wavelength of the incident laser.
In the following, we present a rigorous time domain analysis of the momentum transfer between light and the particle in the different trapping cases.
3. The focused sinusoidal wave light field
Ashkin demonstrated the earliest experiment of capturing microscopic particle in laser trapping by using a focused Gaussian beam within an optical microscope. Since one can build any solutions of the Maxwell’s equations by Fourier analysis with an appropriate combination of the field patterns, which are often referred to as harmonical mode and happen to vary sinusoidally with time, we can study the momentum transfer when a single sinusoidal pulse wave (SSPW) passes through the particle in the trapping space. All the other light field patterns in various laser traps can be considered as superposition of the field patterns of the SSPW. In Fig. 2, we present the light field propagation of a focused negative SSPW, where its wave form curve is shown in Fig. 2(a). The incident light plays a different role with its polarization in laser trapping, so we separate two incident sources of E x and E y polarization by correspondingly adding the incident source to one of transverse electric field components E x or E y in the coordinates shown in Fig. 1, and we refer to the incident electric field component as main electric field component. Fig. 2(b) gives the patterns of the main electric field components E y in sequential time steps. We fix the center of the MTR at the point (x = 685nm, y = 0nm, z = 959nm) in the x-z plane and take the radius of the sphere particle as 0.8μm, and momentum passing through the MTR without the particle is plotted in Fig. 2(c), from which it is obvious that the focused light wave possesses transverse momentum and the different polarization light sets out different momentum. The analytical expression of transverse momentum of focused plane wave can be formulated to compare with these numerical results. Since the MTR is located in the x-z plane, the component S y of the momentum flux is not plotted in Fig. 2(c) because its value is negligible.
In the following we first consider the case of the high refractive index sphere particle (n p > nm, n p = 3.0, nm = 1.33 for water) in the MTR. Fig. 3(a) shows the patterns of the main electric field component E y traveling through the sphere particle in water in sequential time steps. The momentum transfer happens at the interface between the sphere particle and the surrounding medium and at the time when light entered the particle or left the particle. From the simulation, it can be observed that light traveling inside the sphere particle is slowed down due to the higher refractive index of the particle than the surrounding medium. Fig. 3(b) plots the changing of the momentum of different polarization incident as a function of time in the MTR. The fluctuation of the curve in Fig. 3(b) results from the multi-reflection of light at the interface of the sphere particle and the surrounding medium. For ease of comparison, Fig. 4 and Fig. 5 show the main electric field patterns and the curves of the momentum in the MTR for the two cases of the low refractive index and high reflection metal particles, respectively. The low refractive index sphere particle in Fig. 4 is assumed as an air bubble with refractive index of 1.0. The high reflection particle in Fig. 5 is simulated by use of ideal metal model with perfect electric conductor condition  in the surface of the sphere. Figures 3–5 clearly demonstrate various cases of the momentum transfer between light and the different particles in time domain. The total momentum of light among the three cases is calculated by summing the momentum through the MTR in the whole traveling time, which is different in each case due to the various speed of light inside the different dielectric medium. Table 1 presents the total momentum in the three cases and compares them with the total momentum through the MTR without a particle. The total momentum component S x of the light is increased when the SSPW passes the high refractive index sphere particle, which confined the sphere particle to the trapping center on the optical axis. In the cases of the low refractive index particle and the metal particle, the particles are repelled away from the optical axis because the S x of the light is reduced in both cases. These results are consistent with the experimental observations.
4. The optical vortex with a helical phase
Now it has been well known that photons can carry both spin and orbital angular momentum (SAM and OAM). The SAM is associated with polarization of light and the OAM with the azimuthal phase of light. The experimental observations of light-induced rotation by transferring of SAM and OAM from light to microscopic particles have been reported using elliptically polarized laser beams [7, 8] and doughnut beams with helical phase structure , respectively. The optical vortex beam with a phase singularity exp(-ilθ) possesses a helical wavefront, and thus OAM, and also it is referred to as optical doughnut due to its transverse mode of a dark center enclosing a bright ring. Optical vortices may exhibit propagation dynamics similar to hydrodynamic vortex phenomena. To our knowledge, it is still difficult to theoretically describe the transfer between the optical vortex and the microscopic particles. In our simulation, we set a helical phase on an focused incident sinusoidal light wave source enveloped with a Gaussian shape as shown in Fig. 6(a) and its energy density propagation patterns of different topological index l with time are shown in Fig. 6(b). The diameter of the dark center in the optical vortex is enlarged with the topological index l, which can be interpreted as the contribution of the transverse momentum of the optical vortex. Fig. 6(c) shows the dynamics of the energy density distribution of the l = 1 optical vortex varying with time in the focus plane perpendicular to the optical axis. It shows a continuous rotation movement of poynting vector of optical vortex on the transverse plane.
Quantitative analysis of the OAM transfer between optical vortex and the object is required in many scientific fields, such as trapping atom and Bose-Einstein condensate with optical vortex. Theoretical analysis of the OAM transfer from optical vortex and the microscopic particle is difficult in many mathematical approaches. Here we theoretically show the process of the OAM transfer from optical vortex to an object in laser trapping. Whereas the optical doughnut has been experimentally demonstrated to possess the ability to trap the microscopic particles with low refractive index or high reflection, for the sake of brevity, in this paper we only take an example on the high refractive index particle. We set the center of the MTR at the point (685nm, 0nm, 959nm) in the x-z plane and take the high refractive index particle (n p > nm, n p = 3.0, nm = 1.33 for water) with a radius of 0.6μm. Fig. 7(a) shows the main electric field E y varies in sequential time steps when a focused optical vortex beam with the topological index of l = 1 shines on the sphere particle. Momentum of optical vortex with l = 1 and l = 3 passing through the MTR is plotted in the two cases of the particle out of the MTR and in the MTR in Fig. 7(b). The tangential momentum component of the focused OVPB becomes comparable in magnitude with the axial transverse momentum due to the helical wavefront of the optical vortex, and it is found from this figure that the tangential momentum of light is transferred between the optical vortex and the particle. Each momentum component of the charge 3 OVPB passing through the MTR falls down because the greater diameter of the central dark region of the optical vortex with a higher topological order result in decreasing the power shining on the particle. The total momentum passing the MTR fixed at the same position from the optical vortex with the orders l = 1, 2, 3 are compared in Table 2. In Table 2, it can be found that the optical vortex with a topological index l > 1 pushes the particle to rotate around the optical axis, i.e. the direction of OAM of the particle is the same as that of the optical vortex. However, the l = 1 optical vortex propels the particle in the reverse direction and transfers the inverse OAM to the particle, which is different with the traditional experiment demonstration using the optical vortex beam with l > 1 (e.g. l = 3 generally). We investigated the simulation of OAM transfer of Gaussian OVPB with l = 1 in the MTRs, which centers are fixed at the three points (±685nm, 0nm, 959nm) and (0nm, 685nm, 959nm), respectively. Momentum of light passing the MTRs fixed the different locations above is listed in Table 3 and compared between two cases of with and without the sphere particle in the MRT. From the comparison, the tangential momentum difference leads light to pull the sphere to rotate around the optical axis. The rotation movement of the particle due to the OAM transfer from light to the particle can be concluded from this Table. It is worthwhile to note that each vector component of momentum exerted on the particle by linear polarized light is various with the position of the particle, even though the particle moves around the optical axial by control of optical vortex, it also meanwhile fluctuates in different position.
5. Summary and conclusions
We present computer simulations of the FDTD to demonstrate the process of laser trapping. The numerical results are in good agreement with the well-known laser trapping experimental observations. This method offers a practical approach to simulating various optical trapping cases with advantages such as the arbitrary incident source and the arbitrary trapping structure. One can track a particle towards its trap well within light by changing the position of the particle with time in the simulation. The momentum exchange between light and the microscopic particle is studied, and the transverse momentum of the focused light varies with its polarization, and it will be more obvious when light happens to interact with an object. We can exhibit the detailed physical information about the procedure of momentum exchange between light wave and the microscopic particle to interpret each trapping process. Generally in a laser trapping experiment, the trapped particle might be affected by many factors from the environment, for example, Brownian motion and collisions among different particles, therefore this method overcome these drawbacks to provide a pure approach to the particle manipulation induced by laser light. Based on the simulations in this paper, we predict a different OAM transfer by the optical vortex with l = 1, in which the trapped particle is induced to rotate in the inverse direction of OAM of light.
This work was partially supported by the Biomedical Engineering Research Council of Singapore under the Singapore-University of Washington Alliance Programme and the Agency for Science, Technology and Research (A*STAR) of Singapore.
References and links
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