Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam

Efficient Prediction and Analysis of Optical Trapping at Nanoscale via Finite Element Tearing and Interconnecting Method

Numerical simulation plays an important role for the prediction of optical trapping based on plasmonic nano-optical tweezers. However, complicated structures and drastic local field enhancement of plasmonic effects bring great challenges to traditional numerical methods. In this article, an accurate and efficient numerical simulation method based on a dual-primal finite element tearing and interconnecting (FETI-DP) and Maxwell stress tensor is proposed, to calculate the optical force and potential for trapping nanoparticles. A low-rank sparsification approach is introduced to further improve the FETI-DP simulation performance. The proposed method can decompose a large-scale and complex problem into small-scale and simple problems by using non-overlapping domain division and flexible mesh discretization, which exhibits high efficiency and parallelizability. Numerical results show the effectiveness of the proposed method for the prediction and analysis of optical trapping at nanoscale.

Non-conservative optical forces

Undoubtedly, laser tweezers are the most recognized application of optically induced mechanical action. Their operation is usually described in terms of conservative forces originating from intensity gradients. However, the fundamental optical action on matter is non-conservative. We will review different manifestations of non-conservative optical forces (NCF) and discuss their dependence on the specific spatial properties of optical fields that generate them. New developments relevant to the NCF such as tractor beams and transversal forces are also discussed.

Negative optical radiation force and spin torques on subwavelength prolate and oblate spheroids in fractional Bessel-Gauss pincers light-sheets

Fractional Bessel-Gauss light-sheets [J. Opt.19, 055602 (2017)JOOPDB0150-536X10.1088/2040-8986/aa649a], which correspond to finite optical "slices" in 2D and possess asymmetric slit openings and bending characteristics, are examined from the standpoint of optical radiation force and spin torque theories for a subwavelength spheroid with arbitrary orientation in space. The vector angular spectrum decomposition method in addition to the Lorenz gauge condition and Maxwell's equations are used to determine the Cartesian components of the incident radiated electric field of the Bessel-Gauss light-sheets. In the framework of the dipole approximation, the numerical results for the Cartesian components of the optical radiation force and spin torque vectors show that negative forces (oriented in the opposite direction of wave motion) and spin torques arise depending on the beam parameters, the orientation of the subwavelength spheroid in 3D space, and its aspect ratio (i.e., prolate versus oblate). The spin torque sign reversal reveals that counter-clockwise or clockwise rotations around the center of mass of the spheroid can occur. The results find important applications in the application of auto-focusing light-sheets in particle manipulation, rotation, and optical sorting devices.

Computation of radiation pressure force exerted on arbitrary shaped homogeneous particles by high-order Bessel vortex beams using MLFMA

Due to special characteristics of nondiffraction and self reconstruction, the Bessel beams have attracted wide attention in optical trapping and appear to be a dramatic alternative to Gaussian beams. We present in this paper an efficient approach based on the surface integral equations (SIE) to compute the radiation pressure force (RPF) exerted on arbitrary shaped homogeneous particles by high-order Bessel vortex beam (HOBVB). The incident beam is described by vector expressions perfectly satisfy Maxwell's equations. The problem is formulated with the combined tangential formulation (CTF) and solved iteratively with the aid of the multilevel fast multipole algorithm (MLFMA). Then RPF is computed by vector flux of the Maxwell's stress tensor over a spherical surface tightly enclosing the particle and analytical expression for electromagnetic fields of incident beam in near region are used. The numerical predictions are compared with the results of the rigorous method for spherical particle to validate the accuracy of the approach. Some numerical results on relative large particles of complex shape, such as biconcave cell-like particles with different geometry parameters are given, showing powerful capability of our approach. These results are expected to provide useful insights into the RPF exerted on complex shaped particles by HOBVB.

Optical trapping force and torque on spheroidal Rayleigh particles with arbitrary spatial orientations

We investigate the spatial orientation dependence of optical trapping forces and intrinsic torques exerted on spheroidal Rayleigh particles under irradiation of highly focused linearly and circularly polarized beams. It is revealed that the maximal trapping forces and torques strongly depend on the orientation of the spheroid, and the spheroidal particle is driven to be stably trapped at the beam focus with its major axis perpendicular to the optical axis. For a linearly polarized trapping beam, the optical torque is always perpendicular to the plane containing the major axis and the polarization direction of the incident beam. Therefore, the spheroid tends to rotate its major axis along with the polarization direction. However, for a circularly polarized trapping beam, the optical torque is always perpendicular to the plane containing the major axis and the optical axis. What is different from the linear polarization case is that the spheroid tends to have the major axis parallel to the projection of the major axis in the transverse plane. The optical torque in the circular polarization case is half of that in the linear polarization case. These optical trapping properties may be exploited in practical optical manipulation, especially for the nonspherical particle's trapping.

Computational study of radiation torque on arbitrary shaped particles with MLFMA

The surface integral equation (SIE) method is used for the computational study of radiation torque on arbitrarily shaped homogeneous particles. The Multilevel Fast Multipole Algorithm (MLFMA) is employed to reduce memory requirements and improve the capability of SIE. The resultant matrix equations are solved iteratively to obtain equivalent electric and magnetic currents. Then, radiation torque is computed using the vector flux of the pseudotensor over a spherical surface tightly enclosing the particle. We use, therefore, the analytical electromagnetic field expression for incident waves in the near region, instead of the far-field approximation. This avoids the error which may be caused when describing the incident beam. The numerical results of three kinds of non-spherical particles are presented to illustrate the validity and capability of the developed method. It is shown that our method can be applied to predict, in the rigorous sense, the torque from a beam of any shape on a particle of complex configuration with a size parameter as large as 650. The radiation torques on large ellipsoids are exemplified to show the performance of the method and to study the influence that different aspect ratios have on the results. Then, the code is used for the calculation of radiation torque on objects of complex shape including a biconcave cell-like particle and a motor with a non-smooth surface.

Non-spherical gold nanoparticles trapped in optical tweezers: shape matters

We present the results of a theoretical analysis focused on three-dimensional optical trapping of non-spherical gold nanoparticles using a tightly focused laser beam (i.e. optical tweezers). We investigate how the wavelength of the trapping beam enhances trapping stiffness and determines the stable orientation of nonspherical nanoparticles in the optical trap which reveals the optimal trapping wavelength. We consider nanoparticles with diameters being between 20 nm and 254 nm illuminated by a highly focused laser beam at wavelength 1064 nm and compare our results based on the coupled-dipole method with published theoretical and experimental data. We demonstrate that by considering the non-spherical morphology of the nanoparticle we can explain the experimentally observed three-dimensional trapping of plasmonic nanoparticles with size higher than 170 nm. These results will contribute to a better understanding of the trapping and alignment of real metal nanoparticles in optical tweezers and their applications as optically controllable nanosources of heat or probes of weak forces and torques.

Complex rotational dynamics of multiple spheroidal particles in a circularly polarized, dual beam trap

We examine the rotational dynamics of spheroidal particles in an optical trap comprising counter-propagating Gaussian beams of opposing helicity. Isolated spheroids undergo continuous rotation with frequencies determined by their size and aspect ratio, whilst pairs of spheroids display phase locking behaviour. The introduction of additional particles leads to yet more complex behaviour. Experimental results are supported by numerical calculations.

Prediction of metallic nano-optical trapping forces by finite element-boundary integral method

The hybrid of finite element and boundary integral (FE-BI) method is employed to predict nano-optical trapping forces of arbitrarily shaped metallic nanostructures. A preconditioning strategy is proposed to improve the convergence of the iterative solution. Skeletonization is employed to speed up the design and optimization where iteration has to be repeated for each beam configuration. The radiation pressure force (RPF) is computed by vector flux of the Maxwell's stress tensor. Numerical simulations are performed to validate the developed method in analyzing the plasmonic effects as well as the optical trapping forces. It is shown that the proposed method is capable of predicting the trapping forces of complex metallic nanostructures accurately and efficiently.

Optically driven oscillations of ellipsoidal particles. Part II: ray-optics calculations

We report numerical calculations on the mechanical effects of light on micrometer-sized dielectric ellipsoids immersed in water. We used a simple two-dimensional ray-optics model to compute the radiation pressure forces and torques exerted on the object as a function of position and orientation within the laser beam. Integration of the equations of motion, written in the Stokes limit, yields the particle dynamics that we investigated for different aspect ratios k. Whether the beam is collimated or focused, the results show that above a critical aspect ratio k(C), the ellipsoids cannot be stably trapped on the beam axis; the particle never comes to rest and rather oscillates permanently in a back-and-forth motion involving both translation and rotation in the vicinity of the beam. Such oscillations are a direct evidence of the non-conservative character of optical forces. Conversely, stable trapping can be achieved for k < k(C) with the particle standing idle in a vertical position. These predictions are in very good qualitative agreement with experimental observations. The physical origin of the instability may be understood from the force and torque fields whose structures greatly depend on the ellipsoid aspect ratio and beam diameter. The oscillations arise from a non-linear coupling of the forces and torques and the torque amplitude was identified as the bifurcation control parameter. Interestingly, simulations predict that sustained oscillations can be suppressed through the use of two coaxial counterpropagating beams, which may be of interest whenever a static equilibrium is required as in basic force and torque measurements or technological applications.

Intrinsic optical torque of cylindrical vector beams on Rayleigh absorptive spherical particles

The intrinsic optical torque of a focused cylindrical vector beam on a Rayleigh absorptive spherical particle is calculated via the corrected dipole approximation. Numerical results show that, for the radially polarized input field, the torque is distributed in the focal plane strictly along the azimuthal direction anywhere except at the focus. This shows a completely different property from what is observed in the focusing of a circularly polarized beam, where a strong axial torque component arises. For other cylindrically polarized input fields, the torque tends to align itself along the radial direction, as the polarization angle (the angle between the electric vector and the radial direction) changes from 0° to 90°. When limited to considering the torque at the equilibrium position, we find that only for those input fields with polarization angles larger than 50°, the particle experiences a nonzero torque at its equilibrium position. This is verified by showing quantitatively the effects of the polarization angle on the magnitude and orientation of the torque at the equilibrium position.

Rotation, oscillation and hydrodynamic synchronization of optically trapped oblate spheroidal microparticles

While the behavior of optically trapped dielectric spherical particles has been extensively studied, the behavior of non-spherical particles remains mainly unexplored. In this work we focus on the dynamics of oblate spheroidal particles trapped in a tightly focused elliptically-polarized vortex beam. In our experiments we used polystyrene spheroids of aspect ratio of major to minor axes equal to 2.55 and of a volume equal to a sphere of diameter 1.7μm. We demonstrate that such particles can be trapped in three dimensions, with the minor axis oriented perpendicular to both the beam polarization (linear) and the beam propagation, can spin in a circularly polarized beam and an optical vortex beam around the axis parallel with the beam propagation. We also observed that these particles can exhibit a periodic motion in the plane transversal to the beam propagation. We measured that the transfer of the orbital angular momentum from the vortex beam to the spheroid gives rise to torques one order of magnitude stronger comparing to the circularly polarized Gaussian beam. We employed a phase-only spatial light modulator to generate several vortex beam traps with one spheroid in each of them. Due to independent setting of beams parameters we controlled spheroids frequency and sense of rotation and observed hydrodynamic phase and frequency locking of rotating spheroids. These optically driven spheroids offer a simple alternative approach to the former techniques based on birefringent, absorbing or chiral microrotors.

Prediction of radiation pressure force exerted on moving particles by the two-level skeletonization

A fast full-wave method for computing radiation pressure force (RPF) exerted by shaped light beams on moving particles is presented. The problem of evaluating RPF exerted on a moving particle by a single excitation beam is converted into that of computing RPF's exerted on a static particle by multiple beams. The discretization of different beams leads to distinct right hand sides (RHS's) for the matrix system. To avoid solving each RHS by the brute-force manner, the algorithm conducts low-rank decomposition on the excitation matrix consisting of all RHS's to figure out the so-called skeleton light beams by interpolative decomposition (ID). The peak memory requirement of the skeletonization is a bottle-neck if the particle is large. A two-level skeletonization scheme is proposed to solve this problem. Some numerical experiments on arbitrarily shaped homogeneous particles are performed to illustrate the performance and capability of the developed method.

Computation of stress on the surface of a soft homogeneous arbitrarily shaped particle

Prediction of the stress on the surface of an arbitrarily shaped particle of soft material is essential in the study of elastic properties of the particles with optical force. It is also necessary in the manipulation and sorting of small particles with optical tweezers, since a regular-shaped particle, such as a sphere, may be deformed under the nonuniform optical stress on its surface. The stress profile on a spherical or small spheroidal soft particle trapped by shaped beams has been studied, however little work on computing the surface stress of an irregular-shaped particle has been reported. We apply in this paper the surface integral equation with multilevel fast multipole algorithm to compute the surface stress on soft homogeneous arbitrarily shaped particles. The comparison of the computed stress profile with that predicted by the generalized Lorenz-Mie theory for a water droplet of diameter equal to 51 wavelengths in a focused Gaussian beam show that the precision of our method is very good. Then stress profiles on spheroids with different aspect ratios are computed. The particles are illuminated by a Gaussian beam of different waist radius at different incidences. Physical analysis on the mechanism of optical stress is given with help of our recently developed vectorial complex ray model. It is found that the maximum of the stress profile on the surface of prolate spheroids is not only determined by the reflected and refracted rays (orders p=0,1) but also the rays undergoing one or two internal reflections where they focus. Computational study of stress on surface of a biconcave cell-like particle, which is a typical application in life science, is also undertaken.

Computation of radiation pressure force on arbitrary shaped homogenous particles by multilevel fast multipole algorithm

A full-wave numerical method based on the surface integral equation for computing radiation pressure force (RPF) exerted by a shaped light beam on arbitrary shaped homogenous particles is presented. The multilevel fast multipole algorithm is employed to reduce memory requirement and to improve its capability. The resultant matrix equation is solved by using an iterative solver to obtain equivalent electric and magnetic currents. Then RPF is computed by vector flux of the Maxwell's stress tensor over a spherical surface tightly enclosing the particle. So the analytical expressions for electromagnetic fields of incident beam in near region are used. Some numerical results are performed to illustrate the validity and capability of the developed method. Good agreements between our method and the Lorenz-Mie theory for spherical and small spheroidal particle are found while our method has powerful capability for computing RPF of any shaped beam on a relatively large particle of complex shape. Tests for ellipsoidal and red blood cell-like particles illuminated by Gaussian beam have shown that the size of the particle can be as large as 50-100 wavelengths, respectively, for the relative refractive of 1.33 and 1.1.

List of problems for future research in generalized Lorenz-Mie theories and related topics, review and prospectus [invited]

The expression "generalized Lorenz-Mie theories" generically denotes a class of light-scattering theories describing the interaction between an illuminating electromagnetic arbitrary-shaped beam and a particle possessing a high degree of symmetry. This allows one to use the method of separation of variables in which the illuminating beam is expressed as an expansion over a set of basis functions. Such theories have been derived and applied over the past 35 years. Although, as a whole, these theories are now well developed, there remains a list of problems to be solved, some of which are described in this paper.

Calculation of optical forces on an ellipsoid using vectorial ray tracing method

For a triaxial ellipsoid in an optical trap with spherical aberration, the optical forces, torque and stress are analyzed using vectorial ray tracing. The torque will automatically regulate ellipsoid's long axis parallel to optic axis. For a trapped ellipsoid with principal axes in the ratio 1:2:3, the high stress distribution appears in x-z plane. And the optical force at x-axis is weaker than at y-axis due to the shape size. While the ellipsoid departs laterally from trap center, the measurable maximum transverse forces will be weakened due to axial equilibrium and affected by inclined orientation. For an appropriate ring beam, the maximum optical forces are strong in three dimensions, thus, this optical trap is appropriate to trap cells for avoiding damage from laser.

Calculation of radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam

Using the theory of electromagnetic scattering of a uniaxial anisotropic sphere, we derive the analytical expressions of the radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam. The beam's propagation direction is parallel to the primary optical axis of the anisotropic sphere. The effects of the permittivity tensor elements ε(t) and ε(z) on the axial radiation forces are numerically analyzed in detail. The two transverse components of radiation forces exerted on a uniaxial anisotropic sphere, which is distinct from that exerted on an isotropic sphere due to the two eigen waves in the uniaxial anisotropic sphere, are numerically studied as well. The characteristics of the axial and transverse radiation forces are discussed for different radii of the sphere, beam waist width, and distances from the sphere center to the beam center of an off-axis Gaussian beam. The theoretical predictions of radiation forces exerted on a uniaxial anisotropic sphere are hoped to provide effective ways to achieve the improvement of optical tweezers as well as the capture, suspension, and high-precision delivery of anisotropic particles.

Debye series analysis of radiation pressure force exerted on a multilayered sphere

On the basis of generalized Lorenz-Mie theory, the Debye series expansion (DSE) for radiation pressure forces (RPF) exerted on a multilayered sphere induced by focused beams is introduced. The DSE can isolate the contribution of each scattering process to RPF, and give a physical explanation of RPF. Typically, the RPF induced by a Gaussian beam is studied. The DSE is employed to the simulation of RPF corresponding to different scattering processes (diffraction, reflection, refraction, etc.) in detail, and gives the physical mechanism of RPF. The effects of various parameters, such as scattering mode p, beam position, and radius of core for coated spheres, to RPF is researched.

Analytical calculation of axial optical force on a Rayleigh particle illuminated by Gaussian beams beyond the paraxial approximation

We investigate the optical trapping of a Rayleigh particle by a linearly or radially polarized Gaussian beam. The Mie theory is applied to obtain a full solution, with the incident beam being described by the mixed dipole model, which is beyond the paraxial approximation. We then obtain approximate analytical expressions for the optical force, equilibrium position, and trap stiffness for a Rayleigh particle. At equilibrium, the displacement for the particle from the focus scales like a(3) (where a is the radius) for a transparent particle, owing to scattering, whereas for absorptive particles it scales like C+Da(2), owing to absorption. The trap stiffness is found to be proportional to a(3), in agreement with the recent experiment. The radially polarized beam is found to be superior to the linearly polarized beam in the Rayleigh regime in terms of its ability to trap. It is found that the larger the ratio of epsilon(r)/epsilon(i), the closer the equilibrium to the focus, and thus higher stability.

Extreme axial optical force in a standing wave achieved by optimized object shape

Standing wave optical trapping offers many useful advantages in comparison to single beam trapping, especially for submicrometer size particles. It provides axial force stronger by several orders of magnitude, much higher axial trap stiffness, and spatial confinement of particles with higher refractive index. Mainly spherical particles are nowadays considered theoretically and trapped experimentally. In this paper we consider prolate objects of cylindrical symmetry with radius periodically modulated along the axial direction and we present a theoretical study of optimized objects shapes resulting in up to tenfold enhancement of the axial optical force in comparison with the original unmodulated object shape. We obtain analytical formulas for the axial optical force acting on low refractive index objects where the light scattering by the object is negligible. Numerical results based on the coupled dipole method are presented for objects with higher refractive indices and they support the previous simplified analytical conclusions.

Light at work: the use of optical forces for particle manipulation, sorting, and analysis

We review the combinations of optical micro-manipulation with other techniques and their classical and emerging applications to non-contact optical separation and sorting of micro- and nanoparticle suspensions, compositional and structural analysis of specimens, and quantification of force interactions at the microscopic scale. The review aims at inspiring researchers, especially those working outside the optical micro-manipulation field, to find new and interesting applications of these methods.