## New Research In

### Physical Sciences

### Social Sciences

#### Featured Portals

#### Articles by Topic

### Biological Sciences

#### Featured Portals

#### Articles by Topic

- Agricultural Sciences
- Anthropology
- Applied Biological Sciences
- Biochemistry
- Biophysics and Computational Biology
- Cell Biology
- Developmental Biology
- Ecology
- Environmental Sciences
- Evolution
- Genetics
- Immunology and Inflammation
- Medical Sciences
- Microbiology
- Neuroscience
- Pharmacology
- Physiology
- Plant Biology
- Population Biology
- Psychological and Cognitive Sciences
- Sustainability Science
- Systems Biology

# Elliptical orbits of microspheres in an evanescent field

Contributed by Federico Capasso, August 30, 2017 (sent for review January 6, 2017; reviewed by Vincent Daria and Kishan Dholakia)

## 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.

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):**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 (

Due to the symmetries in our system, the trap stiffness, *Height-Dependent Trap Spring Constant*).

In the absence of boundary effects, viscous drag, *Height-Dependent Damping Coefficient*). Due to the spherical symmetry of the particle, in low Reynolds number hydrodynamics (

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 (*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 *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.

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,

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,

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

## 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 **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:*B*.

## 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:

The value of these scattering coefficients was determined using a traditional Mie scattering algorithm, and the infinite series was truncated at a value *A*. 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 (*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,

The motion is separable, with two different cutoff frequencies, **2** implies that though **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-^{∘}): a low-power (*Analysis Flow*.

Fig. 2 plots the predicted and observed 2D trajectories of a 2-

Below both cutoff frequencies, the position of the particle is in phase with the applied force. In this case, the particle’s motion in

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

## 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 *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.

## 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, **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 *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, *Angular Momentum Analysis* and *Comparing the Generation of Angular Momentum with Common Existing Techniques*.

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

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.

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,

## Comparing the Generation of Angular Momentum with Common Existing Techniques

In our setup, we measure an intrinsic angular momentum,

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

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

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. 2*B*). The corner frequencies are given by

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

## 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

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

## Fitting the MSD

The position power spectral density (PSD),

For a Brownian particle in an optical trap driven by an oscillating force at frequency

The damping coefficient

## Net Force on Microparticles for Various Trajectories

Fig. S6 shows the computed net force acting on a 2.2-

## Acknowledgments

We thank the groups of Evelyn Hu and David Weitz at Harvard University for shared equipment and laboratory access. All fabrication was done in the Harvard Center for Nanoscale Sciences (CNS) clean room facility. We acknowledge the support of NSF GFRP Grant DGE1144152 and Research Foundation Flanders Grant 12O9115N.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: capasso{at}seas.harvard.edu or ginis{at}seas.harvard.edu.

Author contributions: L.L., S.K., V.G., and F.C. designed research; L.L., S.K., and A.D.D. performed research; F.C. oversaw research; L.L. and V.G. contributed new reagents/analytic tools; L.L., S.K., V.G., and A.D.D. analyzed data; and L.L., S.K., V.G., A.D.D., and F.C. wrote the paper.

Reviewers: V.D., Australian National University; and K.D., University of St. Andrews.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1714953114/-/DCSupplemental.

Freely available online through the PNAS open access option.

## References

- ↵
- ↵.
- Righini M,
- Volpe G,
- Girard C,
- Petrov D,
- Quidant R

- ↵.
- Min C, et al.

- ↵
- ↵.
- Grigorenko A,
- Roberts N,
- Dickinson M,
- Zhang Y

- ↵
- ↵
- ↵.
- Woolf D,
- Kats MA,
- Capasso F

- ↵.
- Bliokh KY,
- Bekshaev AY,
- Nori F

- ↵
- ↵.
- Rodríguez-Fortuño FJ,
- Engheta N,
- Martínez A,
- Zayats AV

*Nat Commun*6:8799, and erratum (2015) 6:10263. - ↵.
- Hayat A,
- Mueller JPB,
- Capasso F

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵.
- Liu L,
- Kheifets S,
- Ginis V,
- Capasso F

- ↵.
- Leach J,
- Mushfique H,
- di Leonardo R,
- Padgett M,
- Cooper J

- ↵.
- Terray A,
- Oakey J,
- Marr DWM

- ↵
- ↵
- ↵.
- Garces-Chavez V, et al.

- ↵
- ↵
- ↵.
- Wang S,
- Chan CT

- ↵.
- Salary MM,
- Mosallaei H

- ↵
- ↵
- ↵
- ↵.
- Volke-Sepulveda K,
- Garces-Chavez V,
- Chavez-Cerda S,
- Arlt J,
- Dholakia K

- ↵
- ↵
- ↵
- ↵
- ↵.
- Xiao G,
- Yang K,
- Luo H,
- Chen X,
- Xiong W

- ↵
- ↵
- ↵.
- Nieminen TA, et al.

- ↵
- ↵
- ↵.
- Nägele G

- ↵.
- Almaas E,
- Brevik I

- ↵
- ↵.
- Kheifets S,
- Simha A,
- Melin K,
- Li T,
- Raizen MG

- ↵.
- Liu L,
- Woolf A,
- Rodriguez AW,
- Capasso F

- ↵
- ↵.
- Xiao G,
- Yang K,
- Luo H,
- Chen X,
- Xiong W

- ↵.
- Pesce G,
- Volpe G,
- De Luca AC,
- Rusciano G,
- Volpe G

- ↵.
- Bekshaev A,
- Bliokh KY,
- Soskin M

## Citation Manager Formats

## Sign up for Article Alerts

## Article Classifications

- Physical Sciences
- Engineering

## Jump to section

- Article
- Abstract
- Forces on an Optically Driven and Trapped Microsphere
- Height-Dependent Trap Spring Constant
- Height-Dependent Damping Coefficient
- Optical Force on a Dipolar Particle in an Evanescent Field
- Optical Force on a Mie Particle in an Evanescent Field
- Microparticle Mechanical Response
- Experiment
- Analysis Flow
- The Evanescent Wave Technique as a Method of Angular Momentum Generation
- Angular Momentum Analysis
- Comparing the Generation of Angular Momentum with Common Existing Techniques
- Conclusion
- Experimental Set-Up
- Lock-In Algorithm
- Fitting the MSD
- Net Force on Microparticles for Various Trajectories
- Acknowledgments
- Footnotes
- References

- Figures & SI
- Info & Metrics