
Citation: | Liu J, Zheng M, Xiong ZJ, Li ZY. 3D dynamic motion of a dielectric micro-sphere within optical tweezers. Opto-Electron Adv 4, 200015 (2021).. doi: 10.29026/oea.2021.200015 |
Micro-particles trapped in optical tweezers1, 2 can be utilized as carriers or handles that are biochemically linked to biological molecules. With nanometer accuracy of positioning and pico-newton precision of force measurement, optical tweezers offer an ideal platform to exert and measure the forces and torques on single bio-molecules or micro-particles3-5. In 2018, A. Ashkin was awarded the Nobel Physics Prize for his contributions to invent the optical tweezers technology and promote its applications to biological systems. With the rapid development of optical tweezers technology, simple manipulation of particles is no longer sufficient to support either scientific research or industry demands. For example, the first necessary step for statistically analyzing the trajectory of a trapped particle is to track the particle position accurately. This is often achieved by acquiring a video6, 7, stereomicroscopy8, V-shaped micro-mirrors9 of the particle, or other visualization techniques including position sensitive detection10 and quadrant photodiode detection11, which monitors via detecting the backscattering signal or transmitting signal. In fact, these experimental techniques are limited by sampling rate and in 3D imaging applications. Up to now, it is still very hard to track the 3D trajectory of a micro-particle with high precision. Currently, numerical researches12, 13 are conducted to investigate the optical force exerted on trapped particles. However, the detailed dynamics of a particle stepping into a laser trap14, 15 is still an area rarely studied.
This situation inspires us to think of an idea to drive a particle into a steady state by a focused laser beam illuminating it. As it is very difficult or even impossible to determine trapping probabilities through real-world experiments due to the large parameter space (including, for example, the initial states of particles position, velocity, etc.), we have developed a computational framework in which trapping probabilities can be determined easily through numerical simulations. Simulation involves calculating the optical forces and torques, non-optical forces and torques, and combining these components to calculate the overall dynamics of the particle.
A significant concern is how to calculate the optical force and torque of mesoscopic particles. There are two main types of approaches: namely ray optics method and electromagnetic scattering method. In terms of the computational resources and time to solve the dynamic process, the ray optics method has advantages over electromagnetic approaches such as finite-difference time-domain (FDTD) method16 or T-matrix method17, 18. Taking the widely used method of FDTD as an example, the simulation area is needed to be 5 μm × 5 μm × 5 μm for a 3 μm diameter micro-particle. When the micro-particle is far away from the laser trap, the simulation area is even larger. Furthermore, a complete dynamic process involves thousands or even millions of trajectory points, and it is needed to calculate all mechanical quantities (forces and torques) at each point in order to determine the next point of the trajectory by solving electromagnetic fields in a large 3D space. With FDTD or T-matrix method, the time consumption at each point is at the scale of minutes or even hours. Therefore, the solution to a single set of dynamic process costs days or even months. This is a computational burden too heavy which exceeds the capacity of common computers. As for the semi-analytical ray optics method, the time cost is at the scale of millisecond to implement the optical force and torque calculation of each trajectory point, which makes this method a good compromise between clarity and exactness in many situations19, 20. Therefore, we employ the ray optics method to obtain the optical force and torque of the micro-particles in the optical tweezers.
In this paper, we utilize the ray optics theory to calculate optical forces and optical torques. Then, with the influence of viscosity forces and torques taken into account, we numerically analyze the 3D dynamic process of dielectric micro-spheres in optical tweezers on the basis of Newton mechanical equations. In addition, the supplementary videos show the 3D dynamic process of the micro-sphere particles with different initial states of particle positions and velocities. These studies can be valuable to understand experiments that have been performed, to predict the results of potential experiments, or to explore the mechanical dynamics of trapped particles that are not accessible experimentally. Furthermore, these studies can open up opportunities to explore possible applications based on controlling the dynamics of a particle in the optical tweezers.
Investigating the dynamic process of a micro-sphere can be divided into three steps. The first step is to calculate the optical force and torque, the second step is to calculate the non-optical force and torque, and the third and final step is to numerically solve the Newton mechanical equations. This section describes the theory and formulas for solving the dynamic process of the micro-sphere in optical tweezers systematically.
The schematic of the optical tweezers considered in this work and the coordinate system used in our calculation is shown in Fig. 1(a). The center of the optical tweezers, point “O”, is set as the coordinate origin. The collimated Gaussian beam at a wavelength
In general, all the particles experience drag forces coupled with a rapid fluctuation force, which is the result of frequent and numerous collisions with surrounding liquid molecules and the physical origin of famous Brownian motion. The drag force acting on the particle is obtained by the expression
{{\boldsymbol{F}}_{{\rm{drag}}}} = - \gamma \cdot {\boldsymbol{v}}\;, | (1) |
where
\gamma = 6{{\rm{\pi}}} \eta {r_{\rm{s}}}\;, | (2) |
where
{{\boldsymbol{\tau}} _{{\rm{drag}}}} = 8{\rm{\pi}} r_{\rm{s}}^3\eta {\boldsymbol{\omega}} \;, | (3) |
here,
Moreover, the gravity
{\boldsymbol{G}} = - \frac{4}{3}{\rm{\pi}} r_{\rm{s}}^3{\rho _{\rm{s}}}g\;, | (4) |
{{\boldsymbol{F}}_{{\rm{buoy}}}} = \frac{4}{3}{\rm{\pi}} r_{\rm{s}}^3{\rho _{\rm{w}}}g\;. | (5) |
It is noted that the direction of the forces
In order to implement numerical analysis over the dynamic process, the Newton mechanical equations are exploited to accurately model the dynamic parameters of the trapped micro-sphere in the optical tweezers. The equation of the micro-sphere translational motion22-24 can be written as
m\ddot {\boldsymbol{r}} = {{\boldsymbol{F}}_{{\rm{optical}}}}({\boldsymbol{r}},P) + {{\boldsymbol{F}}_{{\rm{drag}}}}({\boldsymbol{v}}) + {\boldsymbol{G}} + {{\boldsymbol{F}}_{{\rm{buoy}}}}\;, | (6) |
where
Similarly, the equation of the micro-sphere rotational motion21 can be expressed by
I\ddot {\bf\textit{ϕ}} = {{\boldsymbol{\tau}} _{{\rm{optical}}}}({\bf\textit{ϕ}} ,P) + {{\boldsymbol{\tau}} _{{\rm{drag}}}}({\boldsymbol{\omega}} )\;, | (7) |
here,
As Eqs. (6) and (7) cannot be solved analytically, the Runge-Kutta methods, a family of implicit and explicit iterative methods, are used in temporal discretization for the approximate solutions of ordinary differential equations (ODEs). One of the above methods, the fourth-order Runge-Kutta method (RK4 method) is utilized. This method has successfully been employed for solving the ODEs25-27. The second-order differential equation Eqs. (6) and (7) each can be converted to two coupled first-order equations respectively,
{\dot {\boldsymbol{v}}} = \frac{{{{\boldsymbol{F}}_{{\rm{optical}}}}({\boldsymbol{r}},P) + {{\boldsymbol{F}}_{{\rm{drag}}}}({\boldsymbol{v}}) + {\boldsymbol{G}} + {{\boldsymbol{F}}_{{\rm{buoy}}}}}}{m} = {\boldsymbol{a}}({\boldsymbol{r}},{\boldsymbol{v}})\;, | (8) |
\dot {\boldsymbol{r}} = {\boldsymbol{v}}\;. | (9) |
And
{\dot{\boldsymbol{\omega}}} = \frac{{{{\boldsymbol{\tau}} _{{\rm{optical}}}}({\bf\textit{ϕ}} ,P) + {{\boldsymbol{\tau}} _{{\rm{drag}}}}({\boldsymbol{\omega}} )}}{I} = {\boldsymbol{\alpha}} ({\bf\textit{ϕ}} ,{\boldsymbol{\omega}} )\;, | (10) |
\dot {\bf\textit{ϕ}} = {\boldsymbol{\omega}} \;. | (11) |
Since the time step has a direct effect on the number of iterations required for running a simulation, we have to choose an appropriate time step size. The characteristic time scale of our entire model is given by the relaxation time
The recursive algorithm for the classical RK4 method can be written as follows. Taking a particular position step
{{\boldsymbol{f}}_{a1}} = {\boldsymbol{a}}({{\boldsymbol{r}}_i},{{\boldsymbol{v}}_i})\;, | (12) |
{{\boldsymbol{f}}_{v1}} = {{\boldsymbol{v}}_i}\;, | (13) |
{{\boldsymbol{f}}_{a2}} = {\boldsymbol{a}}({{\boldsymbol{r}}_i} + \delta t/2 \cdot {{\boldsymbol{f}}_{v1}},{{\boldsymbol{v}}_i} + \delta t/2 \cdot {{\boldsymbol{f}}_{a1}})\;, | (14) |
{{\boldsymbol{f}}_{v2}} = {{\boldsymbol{v}}_i} + \delta t/2 \cdot {{\boldsymbol{f}}_{a1}}\;, | (15) |
{{\boldsymbol{f}}_{a3}} = {\boldsymbol{a}}({{\boldsymbol{r}}_i} + \delta t/2 \cdot {{\boldsymbol{f}}_{v2}},{{\boldsymbol{v}}_i} + \delta t/2 \cdot {{\boldsymbol{f}}_{a2}})\;, | (16) |
{{\boldsymbol{f}}_{v3}} = {{\boldsymbol{v}}_i} + \delta t/2 \cdot {{\boldsymbol{f}}_{a2}}\;, | (17) |
{{\boldsymbol{f}}_{a4}} = {\boldsymbol{a}}({{\boldsymbol{r}}_i} + \delta t/2 \cdot {{\boldsymbol{f}}_{v3}},{{\boldsymbol{v}}_i} + \delta t/2 \cdot {{\boldsymbol{f}}_{a3}})\;, | (18) |
{{\boldsymbol{f}}_{v4}} = {{\boldsymbol{v}}_i} + \delta t/2 \cdot {{\boldsymbol{f}}_{a3}}\;. | (19) |
The position and velocity of the micro-sphere in the next integration time interval are solved as
{{\boldsymbol{r}}_i}_{ + 1} = {{\boldsymbol{r}}_i} + (\delta t/6) \cdot ({{\boldsymbol{f}}_{v1}} + 2{{\boldsymbol{f}}_{v2}} + 2{{\boldsymbol{f}}_{v3}} + {{\boldsymbol{f}}_{v4}})\;, | (20) |
{{\boldsymbol{v}}_i}_{ + 1} = {{\boldsymbol{v}}_i} + (\delta t/6) \cdot ({{\boldsymbol{f}}_{a1}} + 2{{\boldsymbol{f}}_{2a}} + 2{{\boldsymbol{f}}_{a3}} + {{\boldsymbol{f}}_{a4}})\;. | (21) |
Similarly, the mechanical evolution of the remaining physical values
With the above iterative methods, we can obtain the dynamic parameters
On the basis of the above simulation framework, we set a series of initial parameters for the micro-sphere in the optical tweezers, and go further to theoretically draw a picture of its dynamics of mechanical motion. At first, we choose a special initial position of the micro-sphere close to the laser center. This special initiation indicates a high trapping probability, which can help find out whether the simulation framework agrees with experimental results quantitatively or not.
The initial parameters are as follows: the initial position
Having confirmed the accuracy and efficiency of the numerical methods, we proceed to estimate the trapping probabilities by systematically performing simulation at random initial positions and velocities in the parameter space. This greatly helps on judging whether the micro-sphere can be trapped stably and efficiently in the optical tweezers. In the following, we first consider the dynamics of microsphere with different initial positions but with fixed initial zero velocity. We compare the motion trajectories of the micro-sphere when it is set at different initial positions in the optical tweezers. Apart from the initial positions, the rest parameters are as follows, the initial velocity
In the first case, the initial position of the micro-sphere is set to be located on the horizontal plane crossing the trapping center and moving along the x-axis, so that the coordinate on the y- and z- axes is given by
Figure 4 shows the corresponding 3D trajectories of the micro-sphere. The color bar represents the time. This picture can illustrate the temporal evolution of the micro-sphere mechanical motion in 3D space. Moreover, these 3D dynamic processes of the micro-sphere are better visualized in Supporting Information (SI) Video S1. In Fig. 4 and Video S1, it is obvious that the micro-sphere does not follow the shortest straight path to move to the stable position. Besides, the complexity of its trajectory increases as the initial position is located farther away from the laser center. When the micro-sphere is set at
In the second case, the initial position of the micro-sphere is set to be located at the z-axis with
In this part, we consider, calculate, and display the motion trajectories of micro-sphere when it is set at different initial velocities. Similar to the past sections, we have listed the initial parameters, the initial position
The corresponding 3D trajectories and dynamic motions of the micro-sphere at different x-directional initial velocities with
In this part, we go further to investigate the motion trajectories of the micro-sphere in more general initial conditions of positions and velocities. The initial parameters are set as follows: the angular displacement
Figure 7(a) shows the 3D trajectories of the micro-sphere at different z-directional initial velocities with
In the above discussions, we just list several special cases to describe the micro-sphere trapping probabilities. Nevertheless, there will be various combinations of initial parameters to characterize the motions of the micro-sphere in the optical tweezers. In any of these cases, the theoretical and numerical method proposed in this paper can give a reasonable solution and calculation of the mechanical motion dynamic process, and also explain how and why the micro-sphere moves, whether or not, how and why it can be trapped. It should be noted that the initial angular displacement and angular speed have not been discussed or analyzed in the present work. It is mainly because the optical torque is very small as shown in Fig. 1(d)] while the drag torque is relatively very large (
Through simulating the dynamic process, the change of the relevant physical parameters of the micro-sphere [
In summary, we have proposed a method based on semi-analytical ray optics model to visualize the dynamic process of a dielectric micro-sphere in optical tweezers. This approach allows for a fast calculation of optical force and torque exerted upon micro-sphere by focused laser beam with reasonable accuracy and makes it possible to simulate the whole mechanical motion trajectory and process of the particle under Newton’s law. We discuss the trapping probabilities of the micro-sphere at the different initial positions or velocities along the x- and z- axes respectively. It has been found that, unexpectedly, the particle still has the possibility to be trapped by the optical tweezers even if its position is out of the Hookean region. In contrast, the micro-sphere with a large initial velocity will not necessarily be trapped even if its position is in the Hookean region.
On the other hand, even in simple optical tweezers, the dielectric micro-sphere exhibits abundant phases of mechanical motions, including acceleration, deceleration, and turning. Solution and analysis of the 3D microscale particle dynamic process within optical trapping is essential for understanding various mechanisms for engineering the light-matter mechanical interactions, the mechanical motion behavior of molecules or micro-particles in liquid, as well as the biophysics and biochemistry of macromolecules and cells. These dynamics will facilitate the investigation of the single-molecular kinematics, the complex motions of diverse particles and so on. In addition, this method opens up a new way to promote optimal external control in various applications, including molecular winding, nanoparticles sorting, and in vivo manipulations. However, this method is still not mature enough. In the future, we shall perfect the theory of the micro-particle dynamics, and take more factors into consideration, such as the Brownian motion of particles, the change of the viscous drag coefficient (due to the action of the cover glass), the physical characteristics of the particles (shape, radius, refractive index), etc.
This work is supported by the National Natural Science Foundation of China (Grant No. 11974119 and No. 11804399), and the Guangdong Innovative and Entrepreneurial Research Team Program (Grant No. 2016ZT06C594), the Fundamental Research Funds for the Central Universities, South-Central University for Nationalities (Grant No. CZQ20018), and National Key R&D Program of China (No. 2018YFA 0306200).
The authors declare no competing financial interests.
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(a) Schematic of basic optical tweezers. A single laser beam is focused by a high numerical aperture objective lens to form a stable optical trap for microscale particle. (b) and (c) illustrate the calculated optical force acting on a micro-sphere when it moves along the x-axis and z-axis, respectively, in the optical tweezers at the laser power P = 10 mW. Here, the two most important features of optical tweezers are shown: the maximum reverse axial force that characterizes the strength of the trap, and spring constants (the slope of the fit line). (d) and (e) display the calculated optical torque acting on a micro-sphere when it moves along the x-axis and z-axis, respectively.
The dynamic process of the micro-sphere in an optical tweezers. Plots of the (a) position, (b) velocity, (c) angular velocity, and (d) optical force versus time under the initial condition of r0=[0,0,1] μm, v0=[0,0,0] μm/s, ϕ0=[0,0,0] rad, and ω0=[0,0,0] rad/s.
Mechanical motion dynamics of the micro-sphere at different initial positions in the horizontal trapping plane. (a) The x-directional trajectories of the micro-sphere at different initial positions in the x-axis in the optical tweezers with r0x=0.5, 1.0, 1.5, 1.8, 1.9, and 2.0 μm. (b)−(d) display the temporal evolution of the x- and y- and z- directional trajectory, optical force, drag force, and resultant force, respectively, when the micro-sphere is set at the initial position r0x=1.9 μm. (e) and (f) display the temporal evolution of the optical force and the drag force, and the resultant force, respectively, when r0x=2.0 μm.
The 3D trajectories of the micro-sphere in the optical tweezers at different initial positions in the x-axis with r0x=0.5, 1.0, 1.5, 1.8, 1.9, and 2 μm.
Mechanical motion dynamics of the micro-sphere at different initial positions along the optical axis. (a) The z-directional trajectories of the micro-sphere at different initial positions in the z-axis in the optical tweezers with r0x=−2.0, −1.5, −1.0, −0.5, 1.0, 1.4, 1.6, 1.7, 1.75 and 1.9 μm. (b) and (c) The sketches of the optical force and the drag force, and the resultant force over time, respectively, when the micro-sphere is set at the initial position r0x=1.7 μm.
The 3D trajectories of the micro-sphere at different initial velocities in the x-axis in the optical tweezers with v0x=100, 1×104, 1×106, 5×106, 7.5×106, 7.75×106, and 8×106 μm/s.
(a) The 3D trajectories of a micro-sphere at different initial velocities in the z-axis with v0x=1×106, 5×106, 6×106, and 6.5×106 μm/s, when the initial position is r0x=[1,0,0] μm. The plots of (b) the optical force and drag force, and (c) the resultant force over time of the micro-sphere when v0=[6.5 × 106,0,0] μm/s, and r0x=[1,0,0] μm.