Dynamics of microscopic objects in optical tweezers: experimental determination of underdamped regime and numerical simulation using multiscale analysis

Abstract

This article presents new experimental observations and numerical simulations to investigate the dynamic behavior of micro–nano-sized objects under the influence of optical tweezers (OTs). OTs are scientific tools that can apply forces and moments to small particles using a focused laser beam. The motions of three polystyrene microspheres of different diameters, 1,950, 990, and 500 nm, are examined. The results show a transition from the overdamped motion of the largest bead to the underdamped motion of the smallest bead. The experiments are verified using a dynamic model of a microbead under the influence of Gaussian beam OTs that is modeled using ray-optics. The time required to numerically integrate the classic Newton–Euler model is quite long because a picosecond step size must be used. This run time can be reduced using a first-order model, and greatly reduced using a new multiscale model. The difference between these two models is the underdamped behavior predicted by the multiscale model. The experimentally observed underdamped behavior proves that the multiscale model predicts the actual physics of a nano-sized particle moving in a fluid environment characterized by a low Reynolds number.

Appendices

Appendix: Vertical and rotational coordinate

The simulation of the 500 nm bead yields data for all three coordinates, $q_1,\,q_2$, and $q_3$, which can be examined. Here the data for the vertical, $q_2$, and rotational, $q_3$, coordinates are presented for the sake of completeness.

The data for the vertical directions, Fig.16, show gradual movement toward the focal point in the vertical direction. However, the key thing to notice is the oscillations in the vertical direction when the bead reaches the focal line. In addition, there is an offset between the vertical position where the bead settles and the focal point due to optical force cancelation which happens at greater than the focal point [1]. Currently, study is underway to improve experimental bead tracking in the vertical direction to obtain a truer comparison for the simulation.

Fig. 16

Simulation data for $q_2$ the multiscale model 500 nm (CPUtime = $21_{min}$, AbsTol = $10^{-8}$, RelTol = $10^{-7},\,\Delta t = 0.001$ ms)

The rotational coordinate, plotted in Fig.17, shows that there is little rotation as the 500 nm bead approaches the focal line. However, the oscillations at the focal line impart some rotational velocity to the bead. After reaching the focal line, the bead slowly spins, thereby gradually increasing the $q_3$ coordinate. There is no experimental tracking of the bead’s rotation for comparison at this time.

Fig. 17

Simulation data for $q_3$ the multiscale model 500 nm ($\hbox {CPUtime} = 21_{min}$, AbsTol = $10^{-8}$, RelTol = $10^{-7},\,\Delta t = 0.001$ ms)

Forces and moments

1.1 Brownian motion

Random forces and moments in the model, representing Brownian motion, are implemented as Gaussian white noise. They act at and about the mass center of the bead, as shown in Fig. 18. The random forces and moments shown in Fig. 18 representing Brownian motion, are defined, for example, as

$$\begin{aligned}&\mathbf{F}_{\text {Brownian}} = C_{o1}(t) \, \widehat{\mathbf{N}}_1 + C_{o2}(t) \ \widehat{\mathbf{N}}_2 \nonumber \\&\mathbf{T}_{\text {Brownian}} = \bar{L}_S \ C_{o3}(t), \end{aligned}$$

(19)

where $\bar{L}_C$ is a characteristic length of body “S.” The $C_{oi}(t)$ represent forces produced by randomly fluctuating thermal noise. Each component of the random force and that of moment are treated independently as a normally distributed random variable [16]. They have the following expectations, $E[\cdot ]$, or weighted average values:

$$\begin{aligned} E \left[ C_{oi}(t) \right] = \langle \, C_{oi}(t) \, \rangle = 0 = \mu \end{aligned}$$

(20)

and are governed by a fluctuation–dissipation relation expressed as

$$\begin{aligned} E \left[ \ C_{oi}(t_1) \ C_{oj}(t_2) \ \right] = 2 \ \beta \ k_B \ \mathrm{T} \ \delta (t_1-t_2) \delta _{i,j}, \end{aligned}$$

(21)

where $k_B$ and T are the Boltzmann constant and absolute temperature, respectively [15, 16]. The relation in (21) implies that there is no time dependency between the random processes over time; the random sequence of forces does not repeat regularly.

Fig. 18

Brownian Motion for bead

In addition, (20) and (21) imply

$$\begin{aligned} E[ C_{oi}^2(t) ] \ = \ 2 \ \beta \ k_B \ \mathrm{T} \ = \ \mathrm{Var}(C_{oi}(t)) \ = \ \sigma ^2 \end{aligned}$$

(22)

which is the variance of $C_{oi}$. Thus, the $C_{oi}$ can be generated using the Matlab function normrnd($\mu ,\sigma ,\ldots$) which generates random variables with a normal distribution.

The collection of random forces comprise $\varvec{\Gamma }_{\text {Brown}}$. These randomly fluctuating discontinuous functions show numerical integration and so each random variable is held constant during a single integration step; the random variable is updated at the beginning of each step. Thus, the value of each random variable is known before the integration step, and the decomposed value of the random force must equal it. This is accomplished by defining

$$\begin{aligned} \varvec{\Gamma }_{\text {Brown}} = \hbox {R}_{\text {nd}} \ \mathbf{r}_{\text {nd}} = \hbox {R}_{\text {nd}} \ \left[ \begin{array}{l} {C}_{o1} \\ {C}_{o2} \\ {C}_{o3} \\ \vdots \end{array} \right] , \end{aligned}$$

(23)

where $R_{\text {nd}}$ transforms the random forces into generalized active forces. An example of the random forces used is given in Fig. 19.

Fig. 19

Random forces acting on body “S” in Fig. 5 for the 500 nm bead

1.2 Forces

The other forces accounted for are depicted in Fig. 5b and the associated parameters are defined in Table 2. Weight, buoyancy, and drag forces are set as

$$\begin{aligned} \mathbf{F}_{\text {g}}&= -m_S g_0 \ \widehat{\mathbf{N}}_2\end{aligned}$$

(24)

$$\begin{aligned} \mathbf{F}_B&= \rho _{\text {m}} g_0 V_p \ \widehat{\mathbf{N}}_2\end{aligned}$$

(25)

$$\begin{aligned} \mathbf{F}_D&= -\beta _v^{N} \mathbf{V}_{S_0}, \end{aligned}$$

(26)

where

$$\begin{aligned} \beta _v = 6\pi \mu _{\text {m}} r_S \end{aligned}$$

is obtained from Stokes’ Law.

Table 2 Definition of quantities used

The optical forces require more calculations to set. First, the origins of each of the sample rays need to be calculated. To do this, the angle, $\phi _{FS_0}$, between the central ray and $\widehat{N}_2$, as shown in Fig. 20, needs to be found:

$$\begin{aligned} \phi _{FS_0} \ = \ \arctan \frac{q_1}{\acute{f}-q_2}, \end{aligned}$$

(27)

where $\acute{f}$ is the distance between the virtual point $\acute{F}$ and the inertial frame point, $N_{o}$. The origins of the tangential rays, $\rho _{t_1}$, and $\rho _{t_2}$, can then be calculated. First, we calculate the angle, $\theta$, between $FS_{o}$ and the lines connecting $\acute{F}$ to $\rho _{t_1}$ and $\rho _{t_2}$ (see Fig. 20).

$$\begin{aligned}&\sin \theta = \frac{r_S}{\sqrt{q_1^2+(\acute{f}-q_2)^2}}\end{aligned}$$

(28)

$$\begin{aligned}&\cos \theta = \sqrt{1-{\sin }^2 \theta } \end{aligned}$$

(29)

$$\begin{aligned}&\theta = \arctan \frac{\sin \theta }{\cos \theta } \end{aligned}$$

(30)

$$\begin{aligned}&\rho _{t1} = \acute{f} \ \tan (\gamma )\qquad \rho _{t2} = \acute{f} \ \tan (\delta ), \end{aligned}$$

(31)

Fig. 20

Schematic showing ray–bead contact and ray radial origins

where $\gamma = \phi _{FS_0} - \theta$ and $\delta = \phi _{FS_0} + \theta$. Next, since 15 rays are being sampled (of the total number of rays impacting the bead at any given moment), we define the distance, span, between $\rho _{t1}$ and $\rho _{t2}$, and an interval, $\Delta \rho$, between each pair of ray origins, $\rho _k$:

$$\begin{aligned} \rho _k = \rho _{t1} + (k-1) \Delta \rho \qquad k = 1, \cdots , 15, \end{aligned}$$

(32)

where

$$\begin{aligned} \Delta \rho = \frac{\text {span}}{14} \qquad {\text {span}} = \left| \rho _{t2} - \rho _{t1} \right| \end{aligned}$$

(33)

Note that span, as defined above, will begin to approach infinity as $S_0$ approaches a certain distance, $r_S$, from $\acute{F}$. To counteract this, we arbitrarily set

$$\begin{aligned} {\text {span}} = 2R_{\text {obj}} \end{aligned}$$

(34)

Each ray origin will be treated as the x-coordinate (with respect to the inertial reference frame) of the point $P_{k}(\rho _{k},0)$, which is defined as the point of exit of the ray from the objective lens.

1.3 Torques

There are three sources of torque in this system: the torques imparted by the beam, the viscous drag, and Brownian forces.

The moments imparted onto the particle by each of the rays is set as

$$\begin{aligned} \mathbf{M}_{k} = \mathbf{P}_{S_{0} C_{k}} \times \mathbf{F}_{{\text {tot}},k} \end{aligned}$$

(35)

The viscous torque is obtained from

$$\begin{aligned} {T}_{\text {drag}} = - \beta _{\omega } \dot{q}_3, \end{aligned}$$

(36)

where

$$\begin{aligned} \beta _{\omega } = 8\pi \mu _{\text {m}} r_{S}^{3}. \end{aligned}$$

Finally, Brownian moment is,

$$\begin{aligned} {T}_{\text {Brownian}} = \bar{L}_S \ C_{o3}(t), \end{aligned}$$

(37)

where $\bar{L}_S$ is a characteristic length of body ’S’.

1.4 Fresnel coefficients

Before the Fresnel coefficients can be calculated, we first calculate the angle of incidence, $\alpha _k$, and the angle of refraction, $\beta _k$, of each ray within the sphere.

The angles of incidence are calculated using the definitions of the Dot Product and the Cross Product:

$$\begin{aligned} \cos \alpha _k&= \frac{\mathbf{P}_{C_{k}P_k} \cdot \widehat{\mathbf{n}}_k}{|| \mathbf{r}_{C_{k}P_{k}} || || \widehat{\mathbf{n}}_{k} ||} \end{aligned}$$

(38)

$$\begin{aligned} \sin \alpha _k&= \frac{\mathbf{P}_{C_{k}P_{k}} \times \widehat{\mathbf{n}}_k}{|| \mathbf{P}_{C_{k}P_{k}} || || \widehat{\mathbf{n}}_k ||} \end{aligned}$$

(39)

$$\begin{aligned} \alpha _k&= \arctan \frac{\sin \alpha _k}{\cos \alpha _k} \end{aligned}$$

(40)

The angles of refraction are calculated according to Snell’s Law and the Pythagorean Theorem:

$$\begin{aligned} \sin \beta _k&= \frac{n_{m} \sin \alpha _{k}}{n_S} \end{aligned}$$

(41)

$$\begin{aligned} \cos \beta _k&= |\sqrt{1 - \sin ^2 \beta _k}| \end{aligned}$$

(42)

$$\begin{aligned} \beta _k&= \arctan \frac{\sin \beta _k}{\cos \beta _k} \end{aligned}$$

(43)

Based on the above, we calculate the Fresnel reflection and transmission coefficients:

$$\begin{aligned} R_k&= \frac{1}{2} \left[ \frac{n_{\text {m}} \cos \alpha _k - n_S \cos \beta _k}{n_{\text {m}} \cos \alpha _k + n_S \cos \beta _k} ^2 \right. \nonumber \\&\left. +\frac{n_{\text {m}} \cos \beta _k - n_S \cos \alpha _k}{n_{\text {m}} \cos \beta _k + n_S \cos \alpha _k} ^2 \right] \end{aligned}$$

(44)

$$\begin{aligned} T_k&= 1 - R_k \end{aligned}$$

(45)

1.5 Beam model

In the ray-optics approach to modeling OTs, the total light beam is decomposed into individual rays that each propagate in straight lines in a medium of uniform refractive index (see Ref. [1]). Each ray is assigned an appropriate intensity ($I_{\text {ray}}$), direction ($\phi _{\text {ray}}$), and polarization state, and has the characteristics of a plane wave of zero wavelength which can change directions when it reflects, refracts, or changes polarization at dielectric interfaces according to the Fresnel formulas (see Fresnel coefficients section in Appendix). Diffractive effects are neglected in this regime (see Chap. III in Ref. [3]).

The simple ray-optics model of the OTs used here for calculating the optical forces on a sphere of diameter $\gg \lambda$ is illustrated in Fig. 21, adapted from Ref. [1]. The trap consists of an incident parallel beam of arbitrary mode structure and polarization that enters a high numerical-aperture (NA) MO and is focused, ray-by-ray, to a focal line. Computation of the total force imparted to the sphere consists in summing up of the contributions of each ray entering the aperture at radius $\rho _{\text {ray}}$ with respect to the beam axis. The effect of neglecting the finite size of the actual beam focus, which can approach the limit of $\lambda /2n_{m}$ (see Ref. [21]), is negligible for spheres much larger than $\lambda$. The point-focus description of the convergent beam, in which the rays’s directions and momenta continue in straight lines through the focus, gives the correct incident polarization and momentum for each ray. The rays then reflect and refract at the surface of the sphere, giving rise to the optical forces.

Fig. 21

Geometry of an incident ray giving rise to gradient and scattering force contributions, $F_{\text {grad}}$ and $F_{\text {scat}}$. Here, $\theta$ denotes the angle of incidence, $f$ denotes the focal line, and $O$ denotes the sphere’s center of mass

To illustrate the generation of optical forces, consider the force due to a single ray of power $P$ hitting a dielectric sphere at an angle of incidence, $\alpha$, with incident momentum per second $n_{\text {m}} P/c$ (see Figs. 21, 22). The total force imparted onto the sphere by the ray is the sum of the contributions due to the reflected ray of power $PR$ and the infinite number of emergent refracted rays of successively decreasing power $PT^{2},\,PT^{2}R,\,\ldots ,\,PT^{2}R^{n}$, where the quantities $R$ and $T$ are the Fresnel coefficients of reflection and transmission, respectively, of the sphere’s surface at $\alpha$. The net force acting through the sphere’s mass-center, $S_{0}$, can be resolved into $F_{\text {scat}}$ and $F_{\text {grad}}$ components as given by Roosen et al. (see Refs. [30, 31]) (see Force components section in Appendix for a summary of the derivation):

$$\begin{aligned} F_{\text {{scat}}} = \frac{n_{m} P}{c} \left[ \ 1+R\cos 2\alpha - a \ \right] , \end{aligned}$$

(46)

where

$$\begin{aligned} a = \frac{T^{2}\left[ \cos (2\alpha - 2\beta ) - R\cos (2\alpha - \pi ) \right] }{1 + R^{2} - 2R \cos (\pi - 2\beta )} \end{aligned}$$

(47)

and

$$\begin{aligned} F_{{\text {grad}}} = \frac{n_{m} P}{c} \left[ \ R \sin 2 \alpha - \ b \ \right] , \end{aligned}$$

(48)

where

$$\begin{aligned} b = \frac{T^{2}\left[ \sin (2\alpha - 2\beta ) - R\sin (2\alpha - \pi ) \right] }{1 + R^{2} - 2R\cos (\pi - 2\beta )}, \end{aligned}$$

(49)

where $\alpha$ and $\beta$ are the angles of incidence and refraction, respectively. These formulas sum over all the scattered rays, and are therefore exact. The forces are polarization dependent, since $R$ and $T$ are different for rays polarized perpendicular or parallel to the plane of incidence.

Fig. 22

Geometry for calculating the force imparted by a single incident ray. The incident ray has power P and impacts the particle surface at an angle of incidence, $\theta$. Some rays are reflected off (of power PR), while the rest of the rays are infinitely refracted at an angle denoted by r (with portions of the transmitted light passing out of the particle at powers $PT^{2}R^{n}$)

In (46), the $F_{\text {scat}}$ component, pointing in the direction of the incident ray, is denoted as the scattering force component for this single ray (cf Fig. 21), and acts in a direction parallel to the incident ray. Similarly, in (48), the $F_{\text {grad}}$ component, pointing in a direction perpendicular to the ray in the direction of the ray’s axis, is denoted as the gradient force component for the ray (cf Fig. 21). The action of each ray’s gradient force is to pull to particle’s mass-center onto the ray-axis. The net scattering and gradient forces of the whole beam are defined as the vector sums of the scattering and gradient force contributions of each individual ray within the beam. The result is that the particle will be pulled toward the rays of higher power.

1.6 Force frames

The force exerted by a single ray can be resolved into two components: the scattering force, $F_{\text {scat}}$, which acts in the direction of the ray, and the gradient force, $F_{\text {grad}}$, which acts perpendicular to the direction of the ray, in the direction of increasing intensity. In order to accurately model these forces, we first set up the so-called ray-attached frames–axes that correspond to the components of the ray.

First, we calculate the angle each ray is taking (with respect to the line between $N_{0}$ and $F$) using the general formula:

$$\begin{aligned} \phi _k = \arctan \frac{\rho _k}{f}. \end{aligned}$$

(50)

From these angles, formulate the general rotation matrices for each force frame:

$$\begin{aligned} _\mathbf{N}^{\mathbf{F}_k}\mathbf{R} = \left[ \begin{array}{lll} \cos \phi _k &{} - \sin \phi _k &{} 0 \\ \sin \phi _k &{} \cos \phi _k &{} 0 \\ 0 &{} 0 &{} 1 \end{array} \right] . \end{aligned}$$

(51)

These rotation matrices will determine the relationship between the force components in the ray-attached frames and the force components in the inertial reference frame.

1.7 Force components

We start with the equation for the total force exerted by a single ray (see Refs. [1, 30]):

$$\begin{aligned} F_{{\text {tot}}k} = \frac{n_{m} P_k}{c} \left[ 1 + R_{k} e^{2i \alpha _k} - T_{k}^{2} d \ \right] , \end{aligned}$$

(52)

where

$$\begin{aligned} \displaystyle d \ = \ \frac{e^{2i(\alpha _k - \beta _k)}}{1 - R_{k} e^{i(\pi - 2\beta _k)}}. \end{aligned}$$

(53)

In order to eliminate imaginary terms within the denominator (so that later computations are quicker), we first rationalize the fraction in the T-term:

$$\begin{aligned} d&= \frac{e^{i(2\alpha _k - 2\beta _k)}}{1 - R_{k} e^{i(\pi - 2\beta _k)}} \cdot \frac{1 - R_{k} e^{-i(\pi - 2\beta _k)}}{1 - R_{k} e^{-i(\pi - 2\beta _k)}} \end{aligned}$$

(54)

$$\begin{aligned}&= \frac{e^{i(2\alpha _k - 2\beta _k)} \left[ 1 - R_{k} e^{-i(\pi -2\beta )}\right] }{\left[ 1 - R_{k} e^{i(2\alpha _k -2\beta _k)} \right] \left[ 1 - R_{k} e^{-i(2\alpha _k -2\beta _k)} \right] } \nonumber \\&= \frac{e^{i(2\alpha _k -2\beta _k)} - R_{k} e^{\left[ i(2\alpha _k -2\beta _k) -i(\pi -2\beta _k)\right] }}{1 + R_{k}^{2} -R_{k} \left[ e^{i(\pi - 2\beta _k)} + e^{-i(\pi -2\beta _k)} \right] } \nonumber \\&= \frac{e^{i{2\alpha _k - 2\beta _k}} - R_{k} e^{i(2\alpha _k - \pi )}}{1 + R_{k}^{2} - 2R_{k} \cos \left( \pi - 2\beta _k \right) } \end{aligned}$$

(55)

Following Refs. [1, 30], the gradient force, $F_{{\text {grad}},k}$, is taken to be the imaginary component of the total force, while the scattering force, $F_{{\text {scat}},k}$, is taken to be the real component. Using Euler’s Formula, $F_{{\text {tot}},k}$ can be resolved into (46) and (48).

Force vectors can be generated using the ray-attached frames generated earlier:

$$\begin{aligned} \mathbf{F}_{tot,k} = F_{{\text {grad}},k} \widehat{\mathbf{N}}_1 + F_{{\text {scat}},k} \widehat{\mathbf{N}}_2. \end{aligned}$$

(56)

Each force will be applied to a corresponding contact point on the particle, $C_{k}$.