Elliptical orbits of microspheres in an evanescent field

Significance

Using a highly sensitive particle-tracking scheme, we report an observation of elliptical motion of microparticles driven by a single evanescent field. We show that this behavior is highly tunable and predictable using a theoretical model that accounts for Mie scattering and hydrodynamic drag. The significance is twofold. First, our work represents an important step in understanding the detailed dynamics of microparticles in a fluidic environment—especially near a surface—a prerequisite for effective application of optically driven microparticles as functional elements in optofluidic devices. Second, our method could complement current structured-light approaches for inducing orbital motion in microparticles, with readily tunable parameters of motion including orbital frequency, radius, and ellipticity.

Abstract

We examine the motion of periodically driven and optically tweezed microspheres in fluid and find a rich variety of dynamic regimes. We demonstrate, in experiment and in theory, that mean particle motion in 2D is rarely parallel to the direction of the applied force and can even exhibit elliptical orbits with nonzero orbital angular momentum. The behavior is unique in that it depends neither on the nature of the microparticles nor that of the excitation; rather, angular momentum is introduced by the particle’s interaction with the anisotropic fluid and optical trap environment. Overall, we find this motion to be highly tunable and predictable.

Recently, much work has gone into the investigation of optical forces on micro- and nanoparticles near surfaces, primarily in the context of an electric field localized by a microstructured surface (1–5).

Surface-based geometries have raised theoretical excitement due to, for instance, their ability to significantly enhance optical forces (6–8) as well as the emergence of lateral forces due to the extraordinary momentum and spin in evanescent waves (9–12). From an applied point of view, such geometries can enable miniaturization and parallelization of efficient optical traps enabling integration into optofluidic devices (13, 14). Moreover, light-controlled microspheres near surfaces have found applications as force transducers (15–18) and pumps and switches (19, 20), and in general, their collective manipulation is an advancing field (21–23).

However, in the case where a microparticle is within several diameters of a surface, optical and hydrodynamic surface effects cannot be neglected. Effects arising from optical coupling or reflections can complicate trapping and detection schemes (24), and hydrodynamic interactions can cause the motion of a microparticle to become highly nontrivial (25). Despite this, little quantitative study has been done on the dynamics of optically driven particles near a surface. In studies introducing new schemes to optically manipulate matter, the particle’s response function is often ignored (11, 12, 26–29).

In this work, we investigate the dynamics of a system with both near field optical forces and surface-induced hydrodynamic effects (see Fig. 1). By driving the particle with an oscillating force from a modulated evanescent field and tracking the particle’s motion closely in two dimensions, we map out a range of dynamics that can arise from the interplay of optical and hydrodynamic surface effects. We find that the magnitude and direction of the optical force depends on particle size. We also observe that the trajectory of the particle in general does not follow the direction of the force. Instead, the shape and the orientation of the trajectory vary with modulation frequency and distance of the particle from the surface, a result of the anisotropy in both the hydrodynamic drag and the optical trap spring constant.

Fig. 1.

The optically trapped microsphere in its anisotropic environment. The optical trap (660 nm, 12 mW) has a Rayleigh length, $b$, larger than the beam waist, $w$, resulting in a restoring force approximately 5 times stronger in the lateral direction. Additionally, near-wall hydrodynamic coupling results in anisotropic drag that is consistently larger in the vertical direction. As a result, the particle has a different response function in the two directions. The optical force of the pump beam has components in $z$ and $x$. At a driving frequency between the two cutoff frequencies, the particle’s response can have large phase differences between the two directions, generating elliptical orbits. $F_k$ represents the restoring force of the trap. The dissipative drag and the stochastic force are not visualized. For a detailed description of the setup, see Fig. S1.

In particular, we find that under certain conditions, controlled, elliptical motion of a microsphere can be achieved in this configuration, without the use of a beam that carries either orbital or spin angular momentum. This motion is of theoretical and practical interest as it arises due to the particle’s interaction with the inherently anisotropic mechanical environment nearby a solid interface as it moves in response to a periodic driving force. It has several distinguishing features that may make it an applicable technique for generating orbital angular momentum, alongside existing methods predominantly using structured beams carrying orbital or spin angular momentum (30–36). For a detailed discussion, please see The Evanescent Wave Technique as a Method of Angular Momentum Generation, following Experiment.

Forces on an Optically Driven and Trapped Microsphere

An optically trapped microsphere in fluid is well-modeled as a stochastically driven, damped harmonic oscillator in three dimensions (37, 38):

$$m^{\star }\ddot{x_i}(t)=-{\kappa }_{ij}{x}_j(t)-{\gamma }_{ij}\dot{x_j}(t)+B_i(t)+F_i(t).$$ [1]

The terms on the right of Eq. 1 correspond, in order, to the restoring force of the optical trap, the dissipative drag in fluid, the stochastic force due to thermal fluctuations, and an externally applied force. Here, we apply a force and consider motion only in the two measured ($x$ and $z$) dimensions.

Due to the symmetries in our system, the trap stiffness, ${\kappa }_{ij}$, is a diagonal tensor. For a single beam optical trap, its diagonal elements are not equal, ${\kappa }_x\ne {\kappa }_z$; that is, the harmonic potential is anisotropic. Usually, the magnitude of the spring constant in the lateral direction is several times larger than the spring constant in the axial direction (39). In our model describing the particle motion, we approximate that ${\kappa }_i$ is a constant in $z$, an assumption that is generally valid until the separation becomes smaller than the Debye length in water (see Height-Dependent Trap Spring Constant).

In the absence of boundary effects, viscous drag, $\gamma$, is isotropic, given by Stokes’ Law, ${\gamma }_0=6\pi \eta R$, where $\eta$ is the viscosity of the fluid and $R$ is the radius of a sphere. The presence of the boundary results in an anisotropic, height-dependent drag coefficient (40, 41). The predicted drag in the direction perpendicular to the surface ($z$) is always larger than that in the lateral direction ($x$) (see Height-Dependent Damping Coefficient). Due to the spherical symmetry of the particle, in low Reynolds number hydrodynamics ($\mathrm{Re}\ll 1$), we may assume that the drag forces parallel and perpendicular to a plane boundary are uncoupled, such that ${\gamma }_{ij}$ is diagonal with unequal elements (42).

$\overrightarrow{B}(t)$ is the fluctuating thermal force due to collisions between the microparticle and molecules of the fluid. In each dimension, the fluctuations are expected to be uncorrelated, with $⟨B_i(t)⟩=0$, where $\stackrel{\sim }{B}(\omega )$ is spectrally flat. Its mean-squared magnitude is predicted by the fluctuation-dissipation theorem to be $⟨B_i{(t)}^2⟩=2{\gamma }_i{k}_{B}T$, where $k_B$ is the Boltzmann constant. It is important to note that thermal noise increases not only with temperature but also with increased viscous drag and is the dominant source of noise in our system (18).

Lastly, the external force, which drives the periodic motion of the particle, is an optical force produced by the interaction of the microsphere with the evanescent field of a totally internally reflected TM-polarized plane wave. In the dipolar approximation, the interaction can be broadly understood to have two components: one proportional to the gradient of the field, which attracts the particle toward the surface, and the other proportional to the linear momentum of the wave, which pushes the particle along the surface (see Optical Force on a Dipolar Particle in an Evanescent Field).

For small, lossless particles ($\text{Im}[\alpha ]\ll \text{Re}[\alpha ]$), the optical force is predominantly oriented along the z direction. But as the particle increases in size, its interaction with the evanescent wave becomes more complex (43). In this regime, called the Mie regime, light is scattered into every direction in the $xz$ plane. As a result, the direction of the net applied force is observed to rotate toward the horizontal direction. A detailed analysis of the forces acting on the particle in the Mie regime is shown in Optical Force on a Mie Particle in an Evanescent Field.

Height-Dependent Trap Spring Constant

In predicting the trajectories in the main text, the spring constant was assumed to be a constant independent of height. This assumption is roughly valid in the lateral direction but begins to break down in the axial direction near the surface.

At very close separations less than 350 nm from the surface, the spring constant in the axial direction of the beam increases sharply (see Fig. S5). This is due to the particle’s electrostatic repulsion from the surface arising as an additional confining force (discussed in ref. 18). The trap potential narrows and the particle becomes more tightly confined.

Fig. S5.

Fitted trap stiffnesses, ${\kappa }_x$ and ${\kappa }_z$, in the horizontal and vertical directions as a function of height for a 2-$\mu$m-radius PS particle. The line is to guide the eye.

This effect is neglected in the trajectory figures in the main text. For this reason, the trajectory predictions very close to the surface are expected to disagree somewhat from the measured results.

Height-Dependent Damping Coefficient

For slow motion of a microsphere in fluid in close proximity to a hard surface with no-slip boundary conditions, Brenner and Goldman derived expressions for the height-dependent viscous drag in the direction perpendicular to the wall, ${\gamma }_z(z)$ (40), and in the direction parallel to the wall, ${\gamma }_x(z)$ (41), as correction to the Stokes law result ${\gamma }_0=6\pi \eta R$.

The drag is generally larger than the case with no boundary present due to hydrodynamic coupling between the sphere and the wall and generally larger in the perpendicular direction than the parallel direction (see Fig. S4). On contact with the wall, the correction coefficient diverges for the perpendicular direction—that is, ${\gamma }_z\to \mathrm{\infty }$—while the correction coefficient for the parallel direction reaches a constant value ${\gamma }_x\to C$. Far from the wall, the boundary’s influence diminishes and ${\gamma }_x(z)$ and ${\gamma }_z(z)$ approach each other and the isotropic result of Stokes law. The diffusion coefficient depends inversely on the drag as $D(z)=k_{B}T/\gamma (z)$.

Fig. S4.

The measured perpendicular damping coefficient as a function of height above surface $\gamma (z)$ for a PS microsphere is fit to theoretical hydrodynamic predictions (40). The fitted radius is then used to predict the damping coefficient in the lateral direction, which is generally smaller. Fitted results yield a radius of 1.1 $\mu$m for the sphere at a temperature of 22 °C.

Although the height-dependent damping complicates force measurement, since it results in height dependence of the mechanical susceptibility and the thermal noise, we take advantage of the relationship between damping in the two directions to perform the calibration between signal intensity and position in the $x$ direction and to better estimate the $z=0$ point or the point-of-contact in the $z$ direction (18).

Optical Force on a Dipolar Particle in an Evanescent Field

In the dipole approximation, the force on a small, polarizable particle can be calculated analytically. The time-averaged Lorentz force acting upon a dipolar particle with radius $a$ and permittivity ${ϵ}_p$, surrounded by a background with permittivity ${ϵ}_b$, consists of three terms:

$$\overrightarrow{F}=\frac{\text{Re}[\alpha ]}{4}\nabla {\left|\overrightarrow{E_0}\right|}^2+\frac{\text{Im}[\alpha ]}{2}\overrightarrow{k}{\left|\overrightarrow{E_0}\right|}^2-\frac{\text{Im}[\alpha ]}{2}\text{Im}[\overrightarrow{E_0}\nabla \overrightarrow{E_0}],$$ [S4]

where $E_0$ is the amplitude of the electric field with wave vector $k$ and $\alpha$ is the polarizability of the particle in the fluid. For small particles ($ka\ll 1$), this polarizability can be approximated by $\alpha ={\alpha }_0\left(1+2/3\mathrm{i}{k}^3{\left|{\alpha }_0\right|}^2\right)$ with ${\alpha }_0=a^3({ϵ}_r-1)/({ϵ}_r+2)$ and ${ϵ}_r={ϵ}_p/{ϵ}_b$. The first term in Eq. 4 corresponds to the gradient force, whereas the second and the third term are scattering forces. In our setup, we consider an evanescent field generated by an internally reflected TM-polarized plane wave. In this case, the first term pulls the particle toward the surface, the second one pushes the particle along the surface in the direction of propagation, and the final term vanishes.

The ratio of these forces perpendicular to and longitudinal along the surface is thus given by:

$$\left|\frac{F_z}{F_x}\right|=\frac{1\text{Re}[\alpha ]\nabla {\left|\overrightarrow{E_0}\right|}^2}{2\text{Im}[\alpha ]k_x{\left|\overrightarrow{E_0}\right|}^2},$$ [S5]

where $\overrightarrow{E_0}=E_0{\overrightarrow{1}}_E\exp [i{k}_{x}x-qz]$. $|F_z/F_x|$ is thus entirely determined by the ratio of the real and the imaginary parts of the particle’s polarizability $\alpha$ and the evanescent field’s wave vector:

$$\left|\frac{F_z}{F_x}\right|=\frac{q}{k_x}\frac{\text{Re}[\alpha ]}{\text{Im}[\alpha ]}.$$ [S6]

For a given angle of incidence, $q$ and $k_z$ remain fixed. The polarizability $\alpha$ of a small particle is related to its static polarizability ${\alpha }_0$ using Draine’s radiation correction:

$$\alpha ={\alpha }_0{\left(1-\frac{2}{3}\mathrm{i}{k}^3{\alpha }_0\right)}^{-1}\approx {\alpha }_0\left(1+\frac{2}{3}\mathrm{i}{k}^3{\alpha }_0^{\ast }\right).$$ [S7]

For lossless particles, the ratio of the real and the imaginary parts of the polarizability thus equals:

$$\frac{\text{Re}[\alpha ]}{\text{Im}[\alpha ]}={\left(\frac{2}{3}{k}^3{\alpha }_0\right)}^{-1}.$$ [S8]

The ratio of the force in the $z$ and the $x$ direction grows to infinity for decreasing particle radius $a$, as the static polarizability ${\alpha }_0$ is proportional to $a^3$. This effect is shown in the small particle regime $(a<0.1\mu \text{m})$ of Fig. S7B.

Fig. S7.

(A) In the dipole regime ($a<0.1\mu$m), the amplitude of both the lateral and the perpendicular force monotonically increase with increasing particle size. In the Mie regime, the forces continue to grow, with signature Mie oscillations superimposed. (B) The relative amplitude of the attractive force versus the longitudinal force grows to infinity in the dipole regime. In the Mie regime, the forces in both directions are comparable in amplitude.

Optical Force on a Mie Particle in an Evanescent Field

To calculate the optical force on dielectric particles with sizes on the order of the wavelength of light, one needs to solve the Mie scattering problem. As shown in ref. 43, the integral of the Maxwell stress tensor can be expressed as a function of the Mie scattering coefficients. The optical force that acts on a dielectric particle in an evanescent field can thus be evaluated using an algebraic combination of the scattering coefficients. Using this algorithm, the time-averaged attractive force and longitudinal forces was calculated as:

$$\frac{⟨F_z⟩}{a^2{ϵ}_0{E}_0^2}=\mathrm{Re}[\frac{\mathrm{i}{\alpha }^2}{4}\sum _{l=1}^{\mathrm{\infty }}\sum _{m=-l}^l\sqrt{\frac{(l+m+2)(l+m+1)}{(2l+1)(2l+3)}}l(l+2)\times (2n_w^2{a}_{l,m}{a}_{l+1,m+1}^{\ast }+n_w^2{a}_{l,m}{A}_{l+1,m+1}^{\ast }+n_w^2{A}_{l,m}{a}_{l+1,m+1}^{\ast }+2b_{l,m}{b}_{l+1,m+1}^{\ast }+b_{l,m}{B}_{l+1,m+1}^{\ast }+B_{l,m}{b}_{l+1,m+1}^{\ast })+\sqrt{\frac{(l-m+1)(l-m+2)}{(2l+1)(2l+3)}}l(l+2)\times (2n_w^2{a}_{l+1,m-1}{a}_{l,m}^{\ast }+n_w^2{a}_{l+1,m-1}{A}_{l,m}^{\ast }+n_w^2{A}_{l+1,m-1}{a}_{l,m}^{\ast }+2b_{l+1,m-1}{b}_{l,m}^{\ast }+b_{l+1,m-1}{B}_{l,m}^{\ast }+B_{l+1,m-1}{b}_{l,m}^{\ast })-\sqrt{(l+m+1)(l-m)}{n}_w\times (-2a_{l,m}{b}_{l,m+1}^{\ast }+2b_{l,m}{a}_{l,m+1}-a_{l,m}{B}_{l,+1}^{\ast }+b_{l,m}{A}_{l,m+1}^{\ast }+B_{l,m}{a}_{l,m+1}^{\ast }-A_{l,m}{b}_{l,m+1}^{\ast })],$$ [S9]

$$\frac{⟨F_x⟩}{a^2{ϵ}_0{E}_0^2}=\mathrm{Im}[\frac{-{\alpha }^2}{2}\sum _{l=1}^{\mathrm{\infty }}\sum _{m=-l}^l\sqrt{\frac{(l-m+1)(l+m+1)}{(2l+1)(2l+3)}}l(l+2)\times (2n_w^2{a}_{l+1,m}{a}_{l,m}^{\ast }+n_w^2{a}_{l+1,m}{A}_{l,m}^{\ast }+n_w^2{A}_{l+1,m}{a}_{l,m}^{\ast }+2b_{l+1,m}{b}_{l,m}^{\ast }+b_{l+1,m}{B}_{l,m}^{\ast }+B_{l+1,m}{b}_{l,m}^{\ast }+n_{w}m(2a_{l,m}{b}_{l,m}^{\ast }+a_{l,m}{B}_{l,m}^{\ast }+A_{l,m}{b}_{l,m}^{\ast }))].$$ [S10]

The Mie scattering coefficients $A_{l,m}$, $B_{l,m}$, $a_{l,m}$, and $b_{l,m}$ in the previous equation are related to the incident (superscript $(i)$) and the scattered (superscript $(s)$) fields and are defined by:

$$E_r^{(i)}=\frac{a^2{E}_0}{r^2}\sum _{l=1}^{\mathrm{\infty }}\sum _{m=-l}^{l}l(l+1)A_{l,m}{\psi }_l(n_w{k}_{0}r)Y_{l,m}(\theta ,\varphi ),$$ [S11]

$$H_r^{(i)}=\frac{a^2{H}_0}{r^2}\sum _{l=1}^{\mathrm{\infty }}\sum _{m=-l}^{l}l(l+1)B_{l,m}{\psi }_l(n_w{k}_{0}r)Y_{l,m}(\theta ,\varphi ),$$ [S12]

$$E_r^{(s)}=\frac{a^2{E}_0}{r^2}\sum _{l=1}^{\mathrm{\infty }}\sum _{m=-l}^{l}l(l+1)a_{l,m}{\xi }_l^{(1)}(n_w{k}_{0}r)Y_{l,m}(\theta ,\varphi ),$$ [S13]

$$H_r^{(s)}=\frac{a^2{H}_0}{r^2}\sum _{l=1}^{\mathrm{\infty }}\sum _{m=-l}^{l}l(l+1)b_{l,m}{\xi }_l(n_w{k}_{0}r)Y_{l,m}(\theta ,\varphi ),$$ [S14]

where $a$ is the radius of the spherical particle, $Y_{l,m}(\theta ,\varphi )$ are the spherical harmonics, ${\psi }_l(r)=r{j}_l(r)$ with $j_l(r)$ the spherical Bessel function, and ${\xi }_l(r)=r{h}_l^{(1)}(r)$ with $h_l^{(1)}(r)$ the spherical Hankel function of the first kind.

The value of these scattering coefficients was determined using a traditional Mie scattering algorithm, and the infinite series was truncated at a value $l$ for which the relative magnitude of the optical force contribution dropped below ${10}^{-7}$. The result is shown in Fig. S7A. It is interesting to note that in the Mie regime, the longitudinal force along the surface approaches the same amplitude as the attractive force toward the surface. This is clear in the large particle regime ($a>0.5\mu$m) in Fig. S7 B and Inset.

Microparticle Mechanical Response

The mechanical response of our microsphere to an applied force is frequency-dependent and can be found by solving Eq. 1. We can neglect the particle’s inertia while working at time scales much larger than the momentum relaxation time, ${\tau }_p=m^{\star }/\gamma$, which, in our system, is below 1 $\mu$s (44, 45). Inserting a time-harmonic solution, we then find that

$${\stackrel{\sim }{x}}_i(\omega )={\chi }_{ij}(\omega ){\stackrel{\sim }{F}}_j(\omega ),$$ [2]

in which we identified the mechanical susceptibility ${\chi }_{ij}(\omega )$ of the system as ${({\chi }^{-1}(\omega ))}_{ij}={\kappa }_{ij}+i\omega {\gamma }_{ij}$.

The motion is separable, with two different cutoff frequencies, $f_c=\kappa /2\pi \gamma$, in the $x$ and $z$ directions. Eq. 2 implies that though $F_x(t)$ and $F_z(t)$ may be in phase, as is the case with our optically driven system, $x(t)$ and $z(t)$ need not. The maximum phase lag between motion in the two directions is $\pi /2$, making it possible to induce elliptical trajectories of the particle. Furthermore, the linearity of the system ensures that Eq. 2 applies whether the force is random or periodic. Thus, in our case, where the external force on our particle is composed of both a driven and a stochastic component, the thermal motion of the microsphere may be considered separately from the time evolution of the particle’s mean position.

Experiment

A schematic of the experiment is shown in Fig. 1 (also see Fig. S1 for details). All experiments are performed at room temperature in a water-filled, closed, 25-$\mu$m-deep microfluidic chamber. An antireflection (AR) coating is applied to the bottom glass–water interface of the chamber to eliminate standing-wave modulation of the trap laser beam (46). Each polystyrene microsphere is optically trapped by a focused CW laser beam (660-nm-wavelength) via a high numerical aperture (N.A. = 1.2) water-immersion objective. The vertical position of the focus, and consequently the height of the trapped bead, is adjusted using a piezo—by lifting and lowering the objective relative to the chamber. In addition to the trap beam, which enters the chamber from above, two lasers, both $p$-polarized, are incident from below at greater than critical angle (${\theta }_c$ = 61.4^∘): a low-power ($<$1 mW), 637-nm beam acts as the probe for the vertical ($z$) position of the particle (15, 47), and a second laser (785 nm, around 100 mW average power before modulation) generates the periodic optical force that pulls on the microparticle. The second beam, which we refer to as the pump beam, is modulated by a chopper set to a desired frequency. A detailed explanation of the analysis may be found in Analysis Flow.

Fig. S1.

Setup. An optical tweezer is formed by the 660-nm laser beam focused by a high-N.A. water immersion objective (OBJ). The trap confines polystyrene microspheres in water near an AR-coated glass surface; the back-scattered trap beam is used to determine its lateral position via a balanced detector (BAL). The bead-surface separation is controlled by vertical translation of the objective. A low-intensity evanescent wave used for vertical position detection is produced by total internal reflection of a 637-nm (probe) beam. Probe light scattered by the particle is sent to a low-noise photodiode (PD1). A second, high-intensity, 785-nm (pump) evanescent wave exerts an optical force on the particle in the x–z plane. A chopper modulates the intensity of the pump beam, which is monitored by PD2. This signal as well as the signals from BAL and PD1 are digitized and recorded by a computer for lock-in signal processing.

Fig. 2 plots the predicted and observed 2D trajectories of a 2-$\mu$m PS microsphere under the influence of a periodic optical force. Predictions were made based on the optical force, mean trap stiffnesses, and bead radius fitted to experimental data. The measured trajectories are mean trajectories, extracted from 100-s measurements where each point is the averaged 2D position of the particle at a certain phase in the chopper actuation cycle. As the frequency of the modulation is increased, several distinct behaviors are observed that can be understood in relation to the cutoff frequencies in the two spatial directions.

Fig. 2.

(A) Predicted and measured 2D motion of a 1-$\mu$m-radius bead when driven by a periodic optical force from an evanescent field. The blue arrows indicate the direction of the optical force. Each tile in A corresponds to a particular height ($z$) and modulation frequency (${\omega }_0$) and is numbered corresponding to its location on the grid (B). The details of the measurement parameters are given in Table S1. (B) Grid of predicted particle trajectories where the horizontal axis represents the driving frequency and the vertical axis is the height above the surface. The displacement amplitude in nanometers is defined as the length of the semimajor axis of each orbit. Calculated cutoff frequencies in the $x$ and $z$ directions are drawn as blue lines. At all heights, the cutoff frequency in the $z$ direction is lower than that in the $x$ direction, due to a larger trap stiffness in the lateral direction. (C) Measured amplitude and phase of frequency-dependent mechanical susceptibility, $\chi (f)$, at a height of 400 nm, compared with predictions.

Table S1.

Details of experimental parameters in Fig. 2

Fig. 2, no.	Frequency, Hz	Height, nm
1	46	175
2	46	500
3	5	300
4	11	600
5	5	220
6	11	220
7	5	130
8	11	150

Below both cutoff frequencies, the position of the particle is in phase with the applied force. In this case, the particle’s motion in $x$ and $z$ is in phase with one another, and it undergoes linear oscillation with a direction determined primarily by the ratio of the trap stiffnesses ${\kappa }_x/{\kappa }_z$. In our case, the ratio is about 5. Above both cutoff frequencies, the motion is again linear as the oscillations are now out of phase with the applied force in both directions. However, the direction of motion is now determined by the ratio of the damping coefficients ${\gamma }_x/{\gamma }_z$. As the particle nears the surface, ${\gamma }_z$ diverges, while ${\gamma }_x$ approaches a constant value, and the motion becomes more and more parallel to the wall. But between the two cutoff frequencies, the motions in the two spatial directions go out of phase with one another, and stable, elliptical orbits are established.

In Fig. 3 we compare our measurements against Mie theory predictions for a fixed modulation frequency. As the microparticle radius increases, the net optical force from the evanescent field increases for a given field strength and configuration, and the direction of the force rotates slowly toward the horizontal. In addition, since hindered diffusion theory predicts drag near a surface to increase as a function of the ratio $z/R$, where $z$ is the separation and $R$ is the radius, increasing the radius has similar effects as decreasing separation. Therefore, large beads tend to move parallel to the surface, regardless of their direction of excitation.

Fig. 3.

(A) Predicted and measured motion of 0.6- to 2.5-$\mu$m-radius beads when driven by a periodic optical force from an evanescent field that is switched on and off at 20 Hz. The blue arrows indicate the direction of the optical force. Each tile in A corresponds to a particular height and bead radius and is numbered corresponding to its location on the grid (B). The details of the measurement parameters are given in Table S2. (B) Grid of predicted particle trajectories based on Mie theory calculations, where the horizontal axis represents bead radius and the vertical axis is the height above the surface. The displacement amplitude in nanometers is defined as the length of the semimajor axis of each orbit. The trajectories labeled 7 and 8 are for a 2.85-$\mu$m-radius bead, outside the scope of our Mie theory calculations. (C) Ratio of measured optical forces in the $z$ and $x$ directions compared with Mie theory. Shaded area shows the uncertainty in the microsphere index of refraction (n = 1.575 $\pm$ 0.005). Generally, as microparticle radius increases, the direction of the optical force rotates toward the horizontal.

Table S2.

Details of experimental parameters in Fig. 3

Fig. 3, no.	Radius, $\mu$m	Height, nm
1	1.1	150
2	1.1	475
3	1.5	230
4	1.5	650
5	2	280
6	2	560
7	2.8	120
8	2.8	550

Analysis Flow

The unknown parameters in our experiment (calibrated positions, lock-in amplitudes, damping coefficients, and trap stiffnesses in the two spatial directions) are determined sequentially in the procedure diagrammed above. Calibrated positions in $z$ follow a logarithmic relation to scattered light intensity and are determined according to the procedure detailed in ref. 46. In Fig. S2, 0 the parameters are defined, with the known values in black and the unknown values in red. As the analysis progresses, parameters determined in each step turn black in color; in this way, the logical progression of our process is revealed.

Fig. S2.

Analysis flow for determination of unknown parameters: 0 describes what is known at the start of this analysis; 1 and 5 refer to the lock-in algorithm detailed in Lock-In Algorithm; 2 and 4 refer to the best fit to mean-squared-displacement (MSD) of particle position described in Fitting the MSD; and 3 references the fit to hindered diffusion model described in Height-Dependent Damping Coefficient.

The Evanescent Wave Technique as a Method of Angular Momentum Generation

The observed phenomenon may find application as an efficient generator of small-radius orbital motion. While we do not suggest that this method supplants the study and use of structured light beams in colloidal systems, here we provide a detailed quantitative comparison and discuss the distinguishing features of our system.

One manner of quantifying angular motion is to consider the torque applied by the particle to the fluid, ${\tau }_{app}(t)=\overrightarrow{r}(t)\times \gamma \overrightarrow{v}(t)$, where $\gamma$ is the tensor described in Eq. 1. For a particle undergoing steady-state, elliptical motion in fluid, this quantity is on average the same as the torque applied to the particle. Fig. 4 shows ${\tau }_{app}$ accumulating over 100 s for a bead driven at two different frequencies, which agree well with predictions. Another method of quantifying this angular motion is to calculate the constant, nonzero intrinsic orbital angular momentum in the steady state, given by $⟨L_{in}⟩=\omega |\stackrel{\sim }{x}||\stackrel{\sim }{z}|\sin ({\varphi }_z-{\varphi }_x),$ where ${\varphi }_x$ and ${\varphi }_z$ are the phases of the complex mechanical susceptibilities ${\chi }_x$ and ${\chi }_z$ in the $x$ and $z$ directions, respectively, and $⟨L_{in}⟩$ is normalized by mass (see Angular Momentum Analysis). This quantity is useful in allowing direct comparison with existing reports in literature, where we do not have access to values of the damping parameter, $\gamma$, but can estimate velocity and orbital radius. In our unoptimized set-up, the largest $L_{in}$ was measured to be $4.9\times {10}^{-15}{\text{m}}^2/\text{s}$, where ${\varphi }_z-{\varphi }_x$ was closest to $\pi /2$. The configuration for this measurement was a 2-$\mu$m-diameter bead driven at 4.6 Hz at a separation of around 200 nm from the surface (corresponding to point 5 in Fig. 2). In comparing this value with previous reports for dielectric particles driven by structured light, such as by optical vortices (30–33), we find our value comparable when normalized by laser power. With optimization of driving frequency and bead radius or changes in the power or wavelength of the pump beam, further gains may be expected. For details regarding this calculation, see Angular Momentum Analysis and Comparing the Generation of Angular Momentum with Common Existing Techniques.

Fig. 4.

Measured and predicted torque applied to fluid integrated over time for a 2-$\mu$m PS bead driven at 11 Hz and 5 Hz frequency at a height of about 200 nm, corresponding to points 5 and 6 in Fig. 2. The force on the bead is the same in both curves.

It is relevant, at this point, to bring up another body of previous work on the orbital motion of colloids, distinct from structured light studies. One specific scheme involves a particle positioned in between a pair of misaligned beams (48). It is a part of a general class of studies in which particles are observed to undergo circulation while immersed in a static force field with nonzero curl (25, 49). In addition, Angelsky et al. observed orbital angular momentum transfer in circularly polarized Gaussian beams as a manifestation of the macroscopic “spin energy flow,” thus confirming the theory of inhomogeneously polarized paraxial beams (34, 50).

We are careful to distinguish the nature of the orbital motion described in our current report from this class of studies. The motion observed in our work is not fully determined by a particle’s passive interaction with a static optical force field but rather by a periodically driven microsphere’s mechanical response. Since our system exhibits driven rather than passive motion, an array of such trapped particles can be made to move in phase, enabling unique collective behaviors that are thus far unexplored.

In addition, while a particle or a collection of particles may be easily made to orbit in a plane parallel to a nearby surface using existing techniques, methods for obtaining orbits in a plane perpendicular to such a surface are less readily available. Particularly in on-chip geometries where space may be an issue, our method of actuation may provide a useful alternative.

Finally, since the motion observed in our work is actively controlled, main parameters of orbital motion—such as rotational rate and radius—are decoupled and can be independently tuned. For instance, to increase the orbital radius in our experiment without affecting the rotation rate, one can simply raise the pump beam’s intensity without changing its modulation frequency. It should be noted, however, that in our scheme the maximum radius of rotation cannot exceed the limits of the optical trap. If complete flexibility in orbital radii is desired, structured light approaches are more suitable. Additionally, if the goal is near-circular motion, a nearby surface or some other symmetry-breaking element is required, making the work described here potentially unsuitable for manipulation in bulk fluid or in vacuum. Also, larger radii of rotation can be achieved in general by the use of structured light, as the requirement that a particle remains in a single-beam Gaussian harmonic trap may prove too limiting for certain applications.

Thus, our technique can be considered as complementary to existing methods for generating orbital angular momentum, rather than substitutive. The ideas underlying both can be combined to extend the parameter range of orbital angular momentum generation in microparticles using optical forces.

Angular Momentum Analysis

In this section, we discuss how we extract the angular momentum from the experimental data and how we compare it to our analytical model. The intrinsic angular momentum—the angular momentum per unit kilogram—was calculated directly by evaluating $L=\overrightarrow{r}\times \overrightarrow{v}$, both for the experimental data and the analytical model.

However, one needs to be cautious when extracting the mean angular momentum from the experimental data. The differentiation of the displacement to calculate the velocity causes a blue-shift in the noise spectrum, amplifying high-frequency noise. The data, as a result, need to be low pass-filtered to be interpreted correctly. This is achieved using a triangular windowing of the velocity data in the time-domain. The resulting calculated angular momentum as a function of the windowing width is shown in Fig. S8.

Fig. S8.

The angular momentum as a function of the windowing width of the velocity data. At small windowing widths, the angular momentum rapidly drops as the high-frequency noise is filtered out. The angular momentum of the experimental data (blue circles) is extracted by fitting the curve in the region of large windowing widths to the analytical curve of a windowed noiseless ellipse (red curve) and by evaluating that fit at zero windowing width. The result is in excellent agreement with the value predicted from our analytical model (orange curve). (A) Frequency, 4.6 Hz; height, 170 nm. (B) Frequency, 4.6 Hz; height, 230 nm. (C) Frequency, 11 Hz; height, 160 nm. (D) Frequency, 11 Hz; height, 360 nm. The aforementioned results were obtained with the particle having a radius equal to $1.1\mu$m.

In these graphs, we can clearly distinguish different regimes. For low windowing widths (high frequency), noise dominates the signal. As the windowing width increases, this noise is filtered out, but undersampling of the ellipse begins to affect the angular momentum resolved, causing a systematic underestimate of the true angular momentum.

To resolve this issue, we fit the curve in the low-frequency regime—where the high-frequency noise is filtered out—to an equivalently undersampled analytical orbit that does not contain any noise. We then extract the angular momentum of the experimental data as the angular momentum of the fitted ellipse at zero windowing width. As shown in Fig. S8, the predicted values of the angular momenta (red curves) are in excellent agreement with the experimentally fitted ones (yellow curves).

In the main text we provide an analytical estimate for the intrinsic angular momentum expected from such an analysis, which depends only on the angular frequency, $\omega$; amplitudes of oscillation, $|\stackrel{\sim }{x}|$ and $|\stackrel{\sim }{z}|$; and the response phase difference in the two directions, ${\varphi }_z-{\varphi }_x$. A brief derivation is provided below:

$$⟨L/m⟩=⟨\overrightarrow{r}\times \overrightarrow{v}⟩$$ [S15]

where the position and velocity vectors are given by:

$$\overrightarrow{r}=|\stackrel{\sim }{x}|\cos (\omega t-{\varphi }_x)\widehat{x}+|\stackrel{\sim }{z}|\cos (\omega t-{\varphi }_z)\widehat{z},\text{and}\overrightarrow{v}=\frac{d\overrightarrow{r}}{dt}=-|\stackrel{\sim }{x}|\omega \sin (\omega t-{\varphi }_x)\widehat{x}-|\stackrel{\sim }{z}|\omega \sin (\omega t-{\varphi }_z)\widehat{z}.$$ [S16]

Taking their cross-product, simplifying, and taking the time average over each cycle then gives:

$$⟨L/m⟩=\omega |\stackrel{\sim }{x}||\stackrel{\sim }{z}|\sin ({\varphi }_z-{\varphi }_x)$$ [S17]

We note that the calculations performed in this paper assume that the particle motion, together with the gradient of spring and damping constants, at each fixed height are sufficiently small that the parameters $\gamma$, $\kappa$, as well as the amplitude and direction of applied optical force may be treated as constants and the equations of motion considered linear.

Comparing the Generation of Angular Momentum with Common Existing Techniques

In our setup, we measure an intrinsic angular momentum, $L_{in}=\overrightarrow{r}\times \overrightarrow{v}$, of $4.89\times {10}^{-15}{\text{m}}^2/\text{s}$. We cite two results in the literature for comparison (31, 33). The first citation reports using a 700-mW beam to generate rotation with angular frequency 0.07 rad/s; the beads were seen to rotate within an ellipse around 5 μm in radius. Scaling down to the powers used in our experiment (80 mW), the intrinsic angular momentum equals $2.28\times {10}^{-14}{\text{m}}^2/\text{s}$, which is about 4.7 times our result.

The second citation reports particles rotating at 0.06 Hz around an inner ring of a Bessel beam of 2.9-μm radius at a power of 600 mW at the sample surface. Rescaling to our power levels, this intrinsic angular momentum equals $5.64\times {10}^{-14}{\text{m}}^2/\text{s}$, which is about 11 times our result.

Thus, our unoptimized system produced angular momentum in optically driven microparticles within an order of magnitude of previous structured-light results. If desired, the angular momentum generation in our system can be greatly enhanced by performing some simple optimizations. For instance, two orders of magnitude improvement can be expected, if, simply, the pump beam is focused so that the spot size is shrunk by a factor of 3. Currently, the low-intensity beam used for excitation has a semimajor axis of around 100 $\mu$m. Focusing to a spot size of around 30 $\mu$m would increase the field intensity by a factor of 10, and the angular momentum generated by a factor of 100.

In our geometry it is possible to independently tune the radius and angular velocity through the manipulation of the chopping frequency and the power of the pump beam.

The angular frequency at which the particle moves is directly given by the chopping frequency of the pump (evanescent) beam. However, to obtain near circular motion, one needs to operate this frequency between the two corner frequencies that determine the particle’s dynamics (blue lines in Fig. 2B). The corner frequencies are given by $f_{\mathrm{corner}x,z}={\kappa }_{x,z}/{\gamma }_{x,z}$, where ${\kappa }_{x,z}$ and ${\gamma }_{x,z}$ are the spring constants and the damping coefficients in the $x$ and the $z$ direction, respectively. In the vertical ($z$) direction, the corner frequency can be made smaller than 0.5 Hz. This defines a certain lower limit of the chopping, or orbital, frequency. In the longitudinal ($x$) direction, the maximal corner frequency is tunable by adjusting the trap stiffness (most directly by increasing the trap laser power). Although this parameter cannot be increased indefinitely, we can easily tune it such that the upper corner frequency lies above 50 Hz (as shown in Fig. S9). The range of angular rotational rates can thus be adjusted approximately between 0.5 Hz and 50 Hz.

Fig. S9.

PSD in the x direction, where the corner frequency is up to ∼50 Hz. This higher frequency was obtained by increasing the trap stiffness. Raw data (light red), running average (dark red), and predicted PSW (black line) are based on a fit of the MSD.

The radius of the orbital motion, on the other hand, can be controlled by adjusting the power (intensity) of the pumping beam. Indeed, the radius can be estimated as $r=F/(\mathrm{i}\omega \ast \gamma +\kappa )$, where $\omega$ is the angular chopping frequency, $\gamma$ is the damping constant, $\kappa$ is the spring constant, and $F$ has an amplitude proportional to the intensity of the pump beam. This shows that the orbital radius will roughly scale with the pump power.

Conclusion

Using a highly sensitive detection scheme, we investigated the dynamics of trapped microspheres under the influence of a temporally modulated force. We find good agreement between our model and the measured motion validating our method of generating elliptical orbits with sustained nonzero orbital angular momentum.

Experimental Set-Up

For details about other portions of the experimental set-up, including the microfluidic chambers, calibration, AR-coating methods, and imaging optics, please see the supplemental materials of Liu et al. (18).

Lock-In Algorithm

The lock-in algorithm is the same as the one used in ref. 18 and is described in detail in the supplemental materials of that paper and is broadly summarized below.

In general, a lock-in measurement is used to estimate the amplitude and phase of a sinusoidal signal obscured by noise using a reference signal. In this case, the output of the monitor photodiode that records the intensity of the chopper-modulated pump beam serves as the reference signal.

The algorithm generates a sin wave with unity amplitude that is in-phase with the square wave with a fundamental frequency equal to the frequency of modulation.

The noisy (discrete) signal $z_i$ is multiplied by the complex reference signal, $\exp [i{\varphi }_i]$, resulting in the complex lock-in signal $l_i=2z_i\exp [i{\varphi }_i]$.

To correct for the phase delay caused by differences in the signal pathways between the pump monitor photodiode and the TIRM detection photodiode, the delay was directly measured and compensated in the data analysis (18).

The output of the lock-in algorithm corresponds to the amplitude of the signal at the fundamental frequency. The coefficient of the first term of the Fourier series expansion of a square wave is larger than the amplitude of the square wave, by a factor of $\pi /4$. The forces we report in our results corresponds to the amplitude of this fundamental component. It is larger than the amplitude of the square wave but smaller than the peak-to-peak height of the square wave, which corresponds to the DC force of an unmodulated beam, by a factor of $2/\pi$.

Fitting the MSD

The position power spectral density (PSD), $S_x(w)$, pictured in Fig. S3, is defined as the Fourier transform of the autocorrelation function $⟨x(t)x(0)⟩$:

$$S_x(w)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}dt{e}^{i\omega t}⟨x(t)x(0)⟩$$ [S1]

The position autocorrelation function is then related to the MSD by:

$$\text{MSD}=⟨{(x(t)-x(0))}^2⟩=2⟨x^2⟩-2⟨x(t)x(0)⟩$$ [S2]

where $⟨x^2⟩$ is the variance of the function $x(t)$.

Fig. S3.

PSD in the $z$ and $x$ directions. Pictured are raw data, running average, and the predicted PSD based on fit to MSD.

For a Brownian particle in an optical trap driven by an oscillating force at frequency ${\omega }_L$, ignoring inertial effects, the MSD is given by:

$$\mathrm{MSD}(t)=\frac{2k_{B}T}{\kappa }\left[1-e^{-t/{\tau }_c}\right]+{\left(2A\right)}^2\left[1-\cos ({\omega }_{L}t)\right],$$ [S3]

where ${\tau }_c=\gamma /\kappa$.

The damping coefficient $\gamma$ and trap stiffness $\kappa$ are extracted by a least-squares fit to the MSD of the Brownian particle, with lock-in amplitude $A$ and frequency ${\omega }_L$ set to the values determined using the lock-in algorithm described in the previous section. One fit is performed per measurement and per spatial dimension at a given height.

Net Force on Microparticles for Various Trajectories

Fig. S6 shows the computed net force acting on a 2.2-$\mu$m-radius PS microsphere under periodic forcing at various stages of its orbit. The figure shows the physical origin of these elliptical trajectories.

Fig. S6.

Net force normalized by effective mass of particle in nanometers per second squared for a few select trajectories of a 2.2-$\mu$m-radius particle actuated at frequencies between 2 and 100 Hz. Force direction and relative magnitude are plotted on top of particle position at each point in the particle’s orbit.